statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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nhds_within_eq_nhds_within {a : α} {s t u : set α}
(h₀ : a ∈ s) (h₁ : is_open s) (h₂ : t ∩ s = u ∩ s) :
𝓝[t] a = 𝓝[u] a | by rw [nhds_within_restrict t h₀ h₁, nhds_within_restrict u h₀ h₁, h₂] | theorem | nhds_within_eq_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds_within_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_eq_nhds {a : α} {s : set α} : 𝓝[s] a = 𝓝 a ↔ s ∈ 𝓝 a | by rw [nhds_within, inf_eq_left, le_principal_iff] | theorem | nhds_within_eq_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_eq_left",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.nhds_within_eq {a : α} {s : set α} (h : is_open s) (ha : a ∈ s) :
𝓝[s] a = 𝓝 a | nhds_within_eq_nhds.2 $ is_open.mem_nhds h ha | theorem | is_open.nhds_within_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_nhds_within_coinduced {π : α → β} {s : set β} {t : set α} {a : α}
(h : a ∈ t) (ht : is_open t)
(hs : s ∈ @nhds β (topological_space.coinduced (λ x : t, π x) subtype.topological_space) (π a)) :
π ⁻¹' s ∈ 𝓝 a | by { rw ← ht.nhds_within_eq h, exact preimage_nhds_within_coinduced' h ht hs } | lemma | preimage_nhds_within_coinduced | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds",
"preimage_nhds_within_coinduced'",
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_empty (a : α) : 𝓝[∅] a = ⊥ | by rw [nhds_within, principal_empty, inf_bot_eq] | theorem | nhds_within_empty | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_bot_eq",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_union (a : α) (s t : set α) :
𝓝[s ∪ t] a = 𝓝[s] a ⊔ 𝓝[t] a | by { delta nhds_within, rw [←inf_sup_left, sup_principal] } | theorem | nhds_within_union | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_bUnion {ι} {I : set ι} (hI : I.finite) (s : ι → set α) (a : α) :
𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a | set.finite.induction_on hI (by simp) $ λ t T _ _ hT,
by simp only [hT, nhds_within_union, supr_insert, bUnion_insert] | theorem | nhds_within_bUnion | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_union",
"set.finite.induction_on",
"supr_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_sUnion {S : set (set α)} (hS : S.finite) (a : α) :
𝓝[⋃₀ S] a = ⨆ s ∈ S, 𝓝[s] a | by rw [sUnion_eq_bUnion, nhds_within_bUnion hS] | theorem | nhds_within_sUnion | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_bUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_Union {ι} [finite ι] (s : ι → set α) (a : α) : 𝓝[⋃ i, s i] a = ⨆ i, 𝓝[s i] a | by rw [← sUnion_range, nhds_within_sUnion (finite_range s), supr_range] | theorem | nhds_within_Union | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"finite",
"nhds_within_sUnion",
"supr_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_inter (a : α) (s t : set α) :
𝓝[s ∩ t] a = 𝓝[s] a ⊓ 𝓝[t] a | by { delta nhds_within, rw [inf_left_comm, inf_assoc, inf_principal, ←inf_assoc, inf_idem] } | theorem | nhds_within_inter | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_assoc",
"inf_idem",
"inf_left_comm",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_inter' (a : α) (s t : set α) :
𝓝[s ∩ t] a = (𝓝[s] a) ⊓ 𝓟 t | by { delta nhds_within, rw [←inf_principal, inf_assoc] } | theorem | nhds_within_inter' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_assoc",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_inter_of_mem {a : α} {s t : set α} (h : s ∈ 𝓝[t] a) :
𝓝[s ∩ t] a = 𝓝[t] a | by { rw [nhds_within_inter, inf_eq_right], exact nhds_within_le_of_mem h } | theorem | nhds_within_inter_of_mem | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"inf_eq_right",
"nhds_within_inter",
"nhds_within_le_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_inter_of_mem' {a : α} {s t : set α} (h : s ∈ 𝓝[t] a) :
𝓝[t ∩ s] a = 𝓝[t] a | by rw [inter_comm, nhds_within_inter_of_mem h] | theorem | nhds_within_inter_of_mem' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_inter_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_singleton (a : α) : 𝓝[{a}] a = pure a | by rw [nhds_within, principal_singleton, inf_eq_right.2 (pure_le_nhds a)] | theorem | nhds_within_singleton | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within",
"pure_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_insert (a : α) (s : set α) :
𝓝[insert a s] a = pure a ⊔ 𝓝[s] a | by rw [← singleton_union, nhds_within_union, nhds_within_singleton] | theorem | nhds_within_insert | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_singleton",
"nhds_within_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_insert {a : α} {s t : set α} :
t ∈ 𝓝[insert a s] a ↔ a ∈ t ∧ t ∈ 𝓝[s] a | by simp | lemma | mem_nhds_within_insert | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert_mem_nhds_within_insert {a : α} {s t : set α} (h : t ∈ 𝓝[s] a) :
insert a t ∈ 𝓝[insert a s] a | by simp [mem_of_superset h] | lemma | insert_mem_nhds_within_insert | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
insert_mem_nhds_iff {a : α} {s : set α} : insert a s ∈ 𝓝 a ↔ s ∈ 𝓝[≠] a | by simp only [nhds_within, mem_inf_principal, mem_compl_iff, mem_singleton_iff,
or_iff_not_imp_left, insert_def] | lemma | insert_mem_nhds_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within",
"or_iff_not_imp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_compl_singleton_sup_pure (a : α) : 𝓝[≠] a ⊔ pure a = 𝓝 a | by rw [← nhds_within_singleton, ← nhds_within_union, compl_union_self, nhds_within_univ] | theorem | nhds_within_compl_singleton_sup_pure | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_singleton",
"nhds_within_union",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_prod_eq {α : Type*} [topological_space α] {β : Type*} [topological_space β]
(a : α) (b : β) (s : set α) (t : set β) :
𝓝[s ×ˢ t] (a, b) = 𝓝[s] a ×ᶠ 𝓝[t] b | by { delta nhds_within, rw [nhds_prod_eq, ←filter.prod_inf_prod, filter.prod_principal_principal] } | lemma | nhds_within_prod_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter.prod_principal_principal",
"nhds_prod_eq",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_prod {α : Type*} [topological_space α] {β : Type*} [topological_space β]
{s u : set α} {t v : set β} {a : α} {b : β}
(hu : u ∈ 𝓝[s] a) (hv : v ∈ 𝓝[t] b) :
(u ×ˢ v) ∈ 𝓝[s ×ˢ t] (a, b) | by { rw nhds_within_prod_eq, exact prod_mem_prod hu hv, } | lemma | nhds_within_prod | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_prod_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pi_eq' {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
{I : set ι} (hI : I.finite) (s : Π i, set (α i)) (x : Π i, α i) :
𝓝[pi I s] x = ⨅ i, comap (λ x, x i) (𝓝 (x i) ⊓ ⨅ (hi : i ∈ I), 𝓟 (s i)) | by simp only [nhds_within, nhds_pi, filter.pi, comap_inf, comap_infi, pi_def, comap_principal,
← infi_principal_finite hI, ← infi_inf_eq] | lemma | nhds_within_pi_eq' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter.pi",
"infi_inf_eq",
"nhds_pi",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pi_eq {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
{I : set ι} (hI : I.finite) (s : Π i, set (α i)) (x : Π i, α i) :
𝓝[pi I s] x = (⨅ i ∈ I, comap (λ x, x i) (𝓝[s i] (x i))) ⊓
⨅ (i ∉ I), comap (λ x, x i) (𝓝 (x i)) | begin
simp only [nhds_within, nhds_pi, filter.pi, pi_def, ← infi_principal_finite hI, comap_inf,
comap_principal, eval],
rw [infi_split _ (λ i, i ∈ I), inf_right_comm],
simp only [infi_inf_eq]
end | lemma | nhds_within_pi_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter.pi",
"inf_right_comm",
"infi_inf_eq",
"infi_split",
"nhds_pi",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pi_univ_eq {ι : Type*} {α : ι → Type*} [finite ι] [Π i, topological_space (α i)]
(s : Π i, set (α i)) (x : Π i, α i) :
𝓝[pi univ s] x = ⨅ i, comap (λ x, x i) 𝓝[s i] (x i) | by simpa [nhds_within] using nhds_within_pi_eq finite_univ s x | lemma | nhds_within_pi_univ_eq | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"finite",
"nhds_within",
"nhds_within_pi_eq",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pi_eq_bot {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
{I : set ι} {s : Π i, set (α i)} {x : Π i, α i} :
𝓝[pi I s] x = ⊥ ↔ ∃ i ∈ I, 𝓝[s i] (x i) = ⊥ | by simp only [nhds_within, nhds_pi, pi_inf_principal_pi_eq_bot] | lemma | nhds_within_pi_eq_bot | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_pi",
"nhds_within",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_pi_ne_bot {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
{I : set ι} {s : Π i, set (α i)} {x : Π i, α i} :
(𝓝[pi I s] x).ne_bot ↔ ∀ i ∈ I, (𝓝[s i] (x i)).ne_bot | by simp [ne_bot_iff, nhds_within_pi_eq_bot] | lemma | nhds_within_pi_ne_bot | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_within_pi_eq_bot",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.piecewise_nhds_within {f g : α → β} {t : set α} [∀ x, decidable (x ∈ t)]
{a : α} {s : set α} {l : filter β}
(h₀ : tendsto f (𝓝[s ∩ t] a) l) (h₁ : tendsto g (𝓝[s ∩ tᶜ] a) l) :
tendsto (piecewise t f g) (𝓝[s] a) l | by apply tendsto.piecewise; rwa ← nhds_within_inter' | theorem | filter.tendsto.piecewise_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"nhds_within_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.if_nhds_within {f g : α → β} {p : α → Prop} [decidable_pred p]
{a : α} {s : set α} {l : filter β}
(h₀ : tendsto f (𝓝[s ∩ {x | p x}] a) l)
(h₁ : tendsto g (𝓝[s ∩ {x | ¬ p x}] a) l) :
tendsto (λ x, if p x then f x else g x) (𝓝[s] a) l | h₀.piecewise_nhds_within h₁ | theorem | filter.tendsto.if_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nhds_within (f : α → β) (a : α) (s : set α) :
map f (𝓝[s] a) = ⨅ t ∈ {t : set α | a ∈ t ∧ is_open t}, 𝓟 (f '' (t ∩ s)) | ((nhds_within_basis_open a s).map f).eq_binfi | lemma | map_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_open",
"nhds_within_basis_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_mono_left {f : α → β} {a : α}
{s t : set α} {l : filter β} (hst : s ⊆ t) (h : tendsto f (𝓝[t] a) l) :
tendsto f (𝓝[s] a) l | h.mono_left $ nhds_within_mono a hst | theorem | tendsto_nhds_within_mono_left | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_mono_right {f : β → α} {l : filter β}
{a : α} {s t : set α} (hst : s ⊆ t) (h : tendsto f l (𝓝[s] a)) :
tendsto f l (𝓝[t] a) | h.mono_right (nhds_within_mono a hst) | theorem | tendsto_nhds_within_mono_right | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_of_tendsto_nhds {f : α → β} {a : α}
{s : set α} {l : filter β} (h : tendsto f (𝓝 a) l) :
tendsto f (𝓝[s] a) l | h.mono_left inf_le_left | theorem | tendsto_nhds_within_of_tendsto_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_mem_of_tendsto_nhds_within {f : β → α} {a : α}
{s : set α} {l : filter β} (h : tendsto f l (𝓝[s] a)) :
∀ᶠ i in l, f i ∈ s | begin
simp_rw [nhds_within_eq, tendsto_infi, mem_set_of_eq, tendsto_principal, mem_inter_iff,
eventually_and] at h,
exact (h univ ⟨mem_univ a, is_open_univ⟩).2,
end | lemma | eventually_mem_of_tendsto_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"nhds_within_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_of_tendsto_nhds_within {f : β → α} {a : α}
{s : set α} {l : filter β} (h : tendsto f l (𝓝[s] a)) :
tendsto f l (𝓝 a) | h.mono_right nhds_within_le_nhds | lemma | tendsto_nhds_of_tendsto_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
principal_subtype {α : Type*} (s : set α) (t : set {x // x ∈ s}) :
𝓟 t = comap coe (𝓟 ((coe : s → α) '' t)) | by rw [comap_principal, set.preimage_image_eq _ subtype.coe_injective] | theorem | principal_subtype | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"set.preimage_image_eq",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_ne_bot_of_mem {s : set α} {x : α} (hx : x ∈ s) :
ne_bot (𝓝[s] x) | mem_closure_iff_nhds_within_ne_bot.1 $ subset_closure hx | lemma | nhds_within_ne_bot_of_mem | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.mem_of_nhds_within_ne_bot {s : set α} (hs : is_closed s)
{x : α} (hx : ne_bot $ 𝓝[s] x) : x ∈ s | by simpa only [hs.closure_eq] using mem_closure_iff_nhds_within_ne_bot.2 hx | lemma | is_closed.mem_of_nhds_within_ne_bot | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.nhds_within_ne_bot {ι : Type*} {f : ι → α} (h : dense_range f) (x : α) :
ne_bot (𝓝[range f] x) | mem_closure_iff_cluster_pt.1 (h x) | lemma | dense_range.nhds_within_ne_bot | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_pi {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
{I : set ι} {s : Π i, set (α i)} {x : Π i, α i} :
x ∈ closure (pi I s) ↔ ∀ i ∈ I, x i ∈ closure (s i) | by simp only [mem_closure_iff_nhds_within_ne_bot, nhds_within_pi_ne_bot] | lemma | mem_closure_pi | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"mem_closure_iff_nhds_within_ne_bot",
"nhds_within_pi_ne_bot",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_pi_set {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)]
(I : set ι) (s : Π i, set (α i)) :
closure (pi I s) = pi I (λ i, closure (s i)) | set.ext $ λ x, mem_closure_pi | lemma | closure_pi_set | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"mem_closure_pi",
"set.ext",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_pi {ι : Type*} {α : ι → Type*} [Π i, topological_space (α i)] {s : Π i, set (α i)}
(I : set ι) (hs : ∀ i ∈ I, dense (s i)) :
dense (pi I s) | by simp only [dense_iff_closure_eq, closure_pi_set,
pi_congr rfl (λ i hi, (hs i hi).closure_eq), pi_univ] | lemma | dense_pi | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure_pi_set",
"dense",
"dense_iff_closure_eq",
"pi_congr",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_nhds_within_iff {f g : α → β} {s : set α} {a : α} :
(f =ᶠ[𝓝[s] a] g) ↔ ∀ᶠ x in 𝓝 a, x ∈ s → f x = g x | mem_inf_principal | lemma | eventually_eq_nhds_within_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_eq_nhds_within_of_eq_on {f g : α → β} {s : set α} {a : α} (h : eq_on f g s) :
f =ᶠ[𝓝[s] a] g | mem_inf_of_right h | lemma | eventually_eq_nhds_within_of_eq_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.eq_on.eventually_eq_nhds_within {f g : α → β} {s : set α} {a : α} (h : eq_on f g s) :
f =ᶠ[𝓝[s] a] g | eventually_eq_nhds_within_of_eq_on h | lemma | set.eq_on.eventually_eq_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"eventually_eq_nhds_within_of_eq_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_congr {f g : α → β} {s : set α} {a : α} {l : filter β}
(hfg : ∀ x ∈ s, f x = g x) (hf : tendsto f (𝓝[s] a) l) : tendsto g (𝓝[s] a) l | (tendsto_congr' $ eventually_eq_nhds_within_of_eq_on hfg).1 hf | lemma | tendsto_nhds_within_congr | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"eventually_eq_nhds_within_of_eq_on",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_of_forall {s : set α} {a : α} {p : α → Prop} (h : ∀ x ∈ s, p x) :
∀ᶠ x in 𝓝[s] a, p x | mem_inf_of_right h | lemma | eventually_nhds_within_of_forall | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_of_tendsto_nhds_of_eventually_within {a : α} {l : filter β}
{s : set α} (f : β → α) (h1 : tendsto f l (𝓝 a)) (h2 : ∀ᶠ x in l, f x ∈ s) :
tendsto f l (𝓝[s] a) | tendsto_inf.2 ⟨h1, tendsto_principal.2 h2⟩ | lemma | tendsto_nhds_within_of_tendsto_nhds_of_eventually_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_iff {a : α} {l : filter β} {s : set α} {f : β → α} :
tendsto f l (𝓝[s] a) ↔ tendsto f l (𝓝 a) ∧ ∀ᶠ n in l, f n ∈ s | ⟨λ h, ⟨tendsto_nhds_of_tendsto_nhds_within h, eventually_mem_of_tendsto_nhds_within h⟩,
λ h, tendsto_nhds_within_of_tendsto_nhds_of_eventually_within _ h.1 h.2⟩ | lemma | tendsto_nhds_within_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"eventually_mem_of_tendsto_nhds_within",
"filter",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_range {a : α} {l : filter β} {f : β → α} :
tendsto f l (𝓝[range f] a) ↔ tendsto f l (𝓝 a) | ⟨λ h, h.mono_right inf_le_left, λ h, tendsto_inf.2
⟨h, tendsto_principal.2 $ eventually_of_forall mem_range_self⟩⟩ | lemma | tendsto_nhds_within_range | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.eq_of_nhds_within {s : set α} {f g : α → β} {a : α}
(h : f =ᶠ[𝓝[s] a] g) (hmem : a ∈ s) : f a = g a | h.self_of_nhds_within hmem | lemma | filter.eventually_eq.eq_of_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_nhds_within_of_eventually_nhds {α : Type*} [topological_space α]
{s : set α} {a : α} {p : α → Prop} (h : ∀ᶠ x in 𝓝 a, p x) :
∀ᶠ x in 𝓝[s] a, p x | mem_nhds_within_of_mem_nhds h | lemma | eventually_nhds_within_of_eventually_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"mem_nhds_within_of_mem_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_within_subtype {s : set α} {a : {x // x ∈ s}} {t u : set {x // x ∈ s}} :
t ∈ 𝓝[u] a ↔ t ∈ comap (coe : s → α) (𝓝[coe '' u] a) | by rw [nhds_within, nhds_subtype, principal_subtype, ←comap_inf, ←nhds_within] | theorem | mem_nhds_within_subtype | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"nhds_subtype",
"nhds_within",
"principal_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_subtype (s : set α) (a : {x // x ∈ s}) (t : set {x // x ∈ s}) :
𝓝[t] a = comap (coe : s → α) (𝓝[coe '' t] a) | filter.ext $ λ u, mem_nhds_within_subtype | theorem | nhds_within_subtype | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter.ext",
"mem_nhds_within_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_eq_map_subtype_coe {s : set α} {a : α} (h : a ∈ s) :
𝓝[s] a = map (coe : s → α) (𝓝 ⟨a, h⟩) | by simpa only [subtype.range_coe] using (embedding_subtype_coe.map_nhds_eq ⟨a, h⟩).symm | theorem | nhds_within_eq_map_subtype_coe | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_subtype_iff_nhds_within {s : set α} {a : s} {t : set s} :
t ∈ 𝓝 a ↔ coe '' t ∈ 𝓝[s] (a : α) | by rw [nhds_within_eq_map_subtype_coe a.coe_prop, mem_map,
preimage_image_eq _ subtype.coe_injective, subtype.coe_eta] | theorem | mem_nhds_subtype_iff_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"mem_map",
"nhds_within_eq_map_subtype_coe",
"subtype.coe_eta",
"subtype.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_coe_mem_nhds_subtype {s t : set α} {a : s} :
coe ⁻¹' t ∈ 𝓝 a ↔ t ∈ 𝓝[s] ↑a | by simp only [mem_nhds_subtype_iff_nhds_within, subtype.image_preimage_coe, inter_mem_iff,
self_mem_nhds_within, and_true] | theorem | preimage_coe_mem_nhds_subtype | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"mem_nhds_subtype_iff_nhds_within",
"self_mem_nhds_within",
"subtype.image_preimage_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_within_iff_subtype {s : set α} {a : α} (h : a ∈ s) (f : α → β) (l : filter β) :
tendsto f (𝓝[s] a) l ↔ tendsto (s.restrict f) (𝓝 ⟨a, h⟩) l | by simp only [tendsto, nhds_within_eq_map_subtype_coe h, filter.map_map, restrict] | theorem | tendsto_nhds_within_iff_subtype | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"filter",
"filter.map_map",
"nhds_within_eq_map_subtype_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at (f : α → β) (s : set α) (x : α) : Prop | tendsto f (𝓝[s] x) (𝓝 (f x)) | def | continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | A function between topological spaces is continuous at a point `x₀` within a subset `s`
if `f x` tends to `f x₀` when `x` tends to `x₀` while staying within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_within_at.tendsto {f : α → β} {s : set α} {x : α}
(h : continuous_within_at f s x) :
tendsto f (𝓝[s] x) (𝓝 (f x)) | h | lemma | continuous_within_at.tendsto | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | If a function is continuous within `s` at `x`, then it tends to `f x` within `s` by definition.
We register this fact for use with the dot notation, especially to use `tendsto.comp` as
`continuous_within_at.comp` will have a different meaning. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on (f : α → β) (s : set α) : Prop | ∀ x ∈ s, continuous_within_at f s x | def | continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | A function between topological spaces is continuous on a subset `s`
when it's continuous at every point of `s` within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.continuous_within_at {f : α → β} {s : set α} {x : α} (hf : continuous_on f s)
(hx : x ∈ s) : continuous_within_at f s x | hf x hx | lemma | continuous_on.continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_univ (f : α → β) (x : α) :
continuous_within_at f set.univ x ↔ continuous_at f x | by rw [continuous_at, continuous_within_at, nhds_within_univ] | theorem | continuous_within_at_univ | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_iff_continuous_at_restrict (f : α → β) {x : α} {s : set α}
(h : x ∈ s) :
continuous_within_at f s x ↔ continuous_at (s.restrict f) ⟨x, h⟩ | tendsto_nhds_within_iff_subtype h f _ | theorem | continuous_within_at_iff_continuous_at_restrict | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_at",
"continuous_within_at",
"tendsto_nhds_within_iff_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.tendsto_nhds_within {f : α → β} {x : α} {s : set α} {t : set β}
(h : continuous_within_at f s x) (ht : maps_to f s t) :
tendsto f (𝓝[s] x) (𝓝[t] (f x)) | tendsto_inf.2 ⟨h, tendsto_principal.2 $ mem_inf_of_right $ mem_principal.2 $ ht⟩ | theorem | continuous_within_at.tendsto_nhds_within | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.tendsto_nhds_within_image {f : α → β} {x : α} {s : set α}
(h : continuous_within_at f s x) :
tendsto f (𝓝[s] x) (𝓝[f '' s] (f x)) | h.tendsto_nhds_within (maps_to_image _ _) | theorem | continuous_within_at.tendsto_nhds_within_image | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.prod_map {f : α → γ} {g : β → δ} {s : set α} {t : set β}
{x : α} {y : β}
(hf : continuous_within_at f s x) (hg : continuous_within_at g t y) :
continuous_within_at (prod.map f g) (s ×ˢ t) (x, y) | begin
unfold continuous_within_at at *,
rw [nhds_within_prod_eq, prod.map, nhds_prod_eq],
exact hf.prod_map hg,
end | lemma | continuous_within_at.prod_map | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_prod_eq",
"nhds_within_prod_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_pi {ι : Type*} {π : ι → Type*} [∀ i, topological_space (π i)]
{f : α → Π i, π i} {s : set α} {x : α} :
continuous_within_at f s x ↔ ∀ i, continuous_within_at (λ y, f y i) s x | tendsto_pi_nhds | lemma | continuous_within_at_pi | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"tendsto_pi_nhds",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_pi {ι : Type*} {π : ι → Type*} [∀ i, topological_space (π i)]
{f : α → Π i, π i} {s : set α} :
continuous_on f s ↔ ∀ i, continuous_on (λ y, f y i) s | ⟨λ h i x hx, tendsto_pi_nhds.1 (h x hx) i, λ h x hx, tendsto_pi_nhds.2 (λ i, h i x hx)⟩ | lemma | continuous_on_pi | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.fin_insert_nth {n} {π : fin (n + 1) → Type*}
[Π i, topological_space (π i)] (i : fin (n + 1)) {f : α → π i} {a : α} {s : set α}
(hf : continuous_within_at f s a)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous_within_at g s a) :
continuous_within_at (λ a, i.insert_nth (f a) (g a)... | hf.fin_insert_nth i hg | lemma | continuous_within_at.fin_insert_nth | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.fin_insert_nth {n} {π : fin (n + 1) → Type*} [Π i, topological_space (π i)]
(i : fin (n + 1)) {f : α → π i} {s : set α} (hf : continuous_on f s)
{g : α → Π j : fin n, π (i.succ_above j)} (hg : continuous_on g s) :
continuous_on (λ a, i.insert_nth (f a) (g a)) s | λ a ha, (hf a ha).fin_insert_nth i (hg a ha) | lemma | continuous_on.fin_insert_nth | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff {f : α → β} {s : set α} :
continuous_on f s ↔ ∀ x ∈ s, ∀ t : set β, is_open t → f x ∈ t → ∃ u, is_open u ∧ x ∈ u ∧
u ∩ s ⊆ f ⁻¹' t | by simp only [continuous_on, continuous_within_at, tendsto_nhds, mem_nhds_within] | theorem | continuous_on_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at",
"is_open",
"mem_nhds_within",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff_continuous_restrict {f : α → β} {s : set α} :
continuous_on f s ↔ continuous (s.restrict f) | begin
rw [continuous_on, continuous_iff_continuous_at], split,
{ rintros h ⟨x, xs⟩,
exact (continuous_within_at_iff_continuous_at_restrict f xs).mp (h x xs) },
intros h x xs,
exact (continuous_within_at_iff_continuous_at_restrict f xs).mpr (h ⟨x, xs⟩)
end | theorem | continuous_on_iff_continuous_restrict | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_iff_continuous_at",
"continuous_on",
"continuous_within_at_iff_continuous_at_restrict"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_iff' {f : α → β} {s : set α} :
continuous_on f s ↔ ∀ t : set β, is_open t → ∃ u, is_open u ∧ f ⁻¹' t ∩ s = u ∩ s | have ∀ t, is_open (s.restrict f ⁻¹' t) ↔ ∃ (u : set α), is_open u ∧ f ⁻¹' t ∩ s = u ∩ s,
begin
intro t,
rw [is_open_induced_iff, set.restrict_eq, set.preimage_comp],
simp only [subtype.preimage_coe_eq_preimage_coe_iff],
split; { rintros ⟨u, ou, useq⟩, exact ⟨u, ou, useq.symm⟩ }
end,
by rw [continuou... | theorem | continuous_on_iff' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_def",
"continuous_on",
"continuous_on_iff_continuous_restrict",
"is_open",
"is_open_induced_iff",
"set.preimage_comp",
"set.restrict_eq",
"subtype.preimage_coe_eq_preimage_coe_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.mono_dom {α β : Type*} {t₁ t₂ : topological_space α} {t₃ : topological_space β}
(h₁ : t₂ ≤ t₁) {s : set α} {f : α → β} (h₂ : @continuous_on α β t₁ t₃ f s) :
@continuous_on α β t₂ t₃ f s | begin
rw continuous_on_iff' at h₂ ⊢,
assume t ht,
rcases h₂ t ht with ⟨u, hu, h'u⟩,
exact ⟨u, h₁ u hu, h'u⟩
end | lemma | continuous_on.mono_dom | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_iff'",
"topological_space"
] | If a function is continuous on a set for some topologies, then it is
continuous on the same set with respect to any finer topology on the source space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on.mono_rng {α β : Type*} {t₁ : topological_space α} {t₂ t₃ : topological_space β}
(h₁ : t₂ ≤ t₃) {s : set α} {f : α → β} (h₂ : @continuous_on α β t₁ t₂ f s) :
@continuous_on α β t₁ t₃ f s | begin
rw continuous_on_iff' at h₂ ⊢,
assume t ht,
exact h₂ t (h₁ t ht)
end | lemma | continuous_on.mono_rng | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_iff'",
"topological_space"
] | If a function is continuous on a set for some topologies, then it is
continuous on the same set with respect to any coarser topology on the target space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_iff_is_closed {f : α → β} {s : set α} :
continuous_on f s ↔ ∀ t : set β, is_closed t → ∃ u, is_closed u ∧ f ⁻¹' t ∩ s = u ∩ s | have ∀ t, is_closed (s.restrict f ⁻¹' t) ↔ ∃ (u : set α), is_closed u ∧ f ⁻¹' t ∩ s = u ∩ s,
begin
intro t,
rw [is_closed_induced_iff, set.restrict_eq, set.preimage_comp],
simp only [subtype.preimage_coe_eq_preimage_coe_iff, eq_comm]
end,
by rw [continuous_on_iff_continuous_restrict, continuous_iff_is_c... | theorem | continuous_on_iff_is_closed | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_iff_is_closed",
"continuous_on",
"continuous_on_iff_continuous_restrict",
"is_closed",
"is_closed_induced_iff",
"set.preimage_comp",
"set.restrict_eq",
"subtype.preimage_coe_eq_preimage_coe_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.prod_map {f : α → γ} {g : β → δ} {s : set α} {t : set β}
(hf : continuous_on f s) (hg : continuous_on g t) :
continuous_on (prod.map f g) (s ×ˢ t) | λ ⟨x, y⟩ ⟨hx, hy⟩, continuous_within_at.prod_map (hf x hx) (hg y hy) | lemma | continuous_on.prod_map | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at.prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_cover_nhds {ι : Sort*} {f : α → β} {s : ι → set α}
(hs : ∀ x : α, ∃ i, s i ∈ 𝓝 x) (hf : ∀ i, continuous_on f (s i)) :
continuous f | continuous_iff_continuous_at.mpr $ λ x, let ⟨i, hi⟩ := hs x in
by { rw [continuous_at, ← nhds_within_eq_nhds.2 hi], exact hf _ _ (mem_of_mem_nhds hi) } | lemma | continuous_of_cover_nhds | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_at",
"continuous_on",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_empty (f : α → β) : continuous_on f ∅ | λ x, false.elim | lemma | continuous_on_empty | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_singleton (f : α → β) (a : α) : continuous_on f {a} | forall_eq.2 $ by simpa only [continuous_within_at, nhds_within_singleton, tendsto_pure_left]
using λ s, mem_of_mem_nhds | lemma | continuous_on_singleton | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_within_at",
"mem_of_mem_nhds",
"nhds_within_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.continuous_on {s : set α} (hs : s.subsingleton) (f : α → β) :
continuous_on f s | hs.induction_on (continuous_on_empty f) (continuous_on_singleton f) | lemma | set.subsingleton.continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_on",
"continuous_on_empty",
"continuous_on_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nhds_within_le_comap {x : α} {s : set α} {f : α → β} (ctsf : continuous_within_at f s x) :
𝓝[s] x ≤ comap f (𝓝[f '' s] (f x)) | ctsf.tendsto_nhds_within_image.le_comap | theorem | nhds_within_le_comap | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_nhds_within_range {α} (f : α → β) (y : β) :
comap f (𝓝[range f] y) = comap f (𝓝 y) | comap_inf_principal_range | lemma | comap_nhds_within_range | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff_continuous_on_univ {f : α → β} : continuous f ↔ continuous_on f univ | by simp [continuous_iff_continuous_at, continuous_on, continuous_at, continuous_within_at,
nhds_within_univ] | lemma | continuous_iff_continuous_on_univ | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous",
"continuous_at",
"continuous_iff_continuous_at",
"continuous_on",
"continuous_within_at",
"nhds_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.mono {f : α → β} {s t : set α} {x : α} (h : continuous_within_at f t x)
(hs : s ⊆ t) : continuous_within_at f s x | h.mono_left (nhds_within_mono x hs) | lemma | continuous_within_at.mono | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.mono_of_mem {f : α → β} {s t : set α} {x : α}
(h : continuous_within_at f t x) (hs : t ∈ 𝓝[s] x) : continuous_within_at f s x | h.mono_left (nhds_within_le_of_mem hs) | lemma | continuous_within_at.mono_of_mem | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_le_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_inter' {f : α → β} {s t : set α} {x : α} (h : t ∈ 𝓝[s] x) :
continuous_within_at f (s ∩ t) x ↔ continuous_within_at f s x | by simp [continuous_within_at, nhds_within_restrict'' s h] | lemma | continuous_within_at_inter' | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_restrict''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_inter {f : α → β} {s t : set α} {x : α} (h : t ∈ 𝓝 x) :
continuous_within_at f (s ∩ t) x ↔ continuous_within_at f s x | by simp [continuous_within_at, nhds_within_restrict' s h] | lemma | continuous_within_at_inter | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_restrict'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_union {f : α → β} {s t : set α} {x : α} :
continuous_within_at f (s ∪ t) x ↔ continuous_within_at f s x ∧ continuous_within_at f t x | by simp only [continuous_within_at, nhds_within_union, tendsto_sup] | lemma | continuous_within_at_union | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.union {f : α → β} {s t : set α} {x : α}
(hs : continuous_within_at f s x) (ht : continuous_within_at f t x) :
continuous_within_at f (s ∪ t) x | continuous_within_at_union.2 ⟨hs, ht⟩ | lemma | continuous_within_at.union | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.mem_closure_image {f : α → β} {s : set α} {x : α}
(h : continuous_within_at f s x) (hx : x ∈ closure s) : f x ∈ closure (f '' s) | by haveI := (mem_closure_iff_nhds_within_ne_bot.1 hx);
exact (mem_closure_of_tendsto h $
mem_of_superset self_mem_nhds_within (subset_preimage_image f s)) | lemma | continuous_within_at.mem_closure_image | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_within_at",
"mem_closure_of_tendsto",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.mem_closure {f : α → β} {s : set α} {x : α} {A : set β}
(h : continuous_within_at f s x) (hx : x ∈ closure s) (hA : maps_to f s A) : f x ∈ closure A | closure_mono (image_subset_iff.2 hA) (h.mem_closure_image hx) | lemma | continuous_within_at.mem_closure | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"closure_mono",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.maps_to.closure_of_continuous_within_at {f : α → β} {s : set α} {t : set β}
(h : maps_to f s t) (hc : ∀ x ∈ closure s, continuous_within_at f s x) :
maps_to f (closure s) (closure t) | λ x hx, (hc x hx).mem_closure hx h | lemma | set.maps_to.closure_of_continuous_within_at | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.maps_to.closure_of_continuous_on {f : α → β} {s : set α} {t : set β}
(h : maps_to f s t) (hc : continuous_on f (closure s)) :
maps_to f (closure s) (closure t) | h.closure_of_continuous_within_at $ λ x hx, (hc x hx).mono subset_closure | lemma | set.maps_to.closure_of_continuous_on | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_on",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.image_closure {f : α → β} {s : set α}
(hf : ∀ x ∈ closure s, continuous_within_at f s x) :
f '' (closure s) ⊆ closure (f '' s) | maps_to'.1 $ (maps_to_image f s).closure_of_continuous_within_at hf | lemma | continuous_within_at.image_closure | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.image_closure {f : α → β} {s : set α} (hf : continuous_on f (closure s)) :
f '' (closure s) ⊆ closure (f '' s) | continuous_within_at.image_closure $ λ x hx, (hf x hx).mono subset_closure | lemma | continuous_on.image_closure | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"closure",
"continuous_on",
"continuous_within_at.image_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_singleton {f : α → β} {x : α} : continuous_within_at f {x} x | by simp only [continuous_within_at, nhds_within_singleton, tendsto_pure_nhds] | lemma | continuous_within_at_singleton | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"nhds_within_singleton",
"tendsto_pure_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_insert_self {f : α → β} {x : α} {s : set α} :
continuous_within_at f (insert x s) x ↔ continuous_within_at f s x | by simp only [← singleton_union, continuous_within_at_union, continuous_within_at_singleton,
true_and] | lemma | continuous_within_at_insert_self | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at",
"continuous_within_at_singleton",
"continuous_within_at_union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.diff_iff {f : α → β} {s t : set α} {x : α}
(ht : continuous_within_at f t x) :
continuous_within_at f (s \ t) x ↔ continuous_within_at f s x | ⟨λ h, (h.union ht).mono $ by simp only [diff_union_self, subset_union_left],
λ h, h.mono (diff_subset _ _)⟩ | lemma | continuous_within_at.diff_iff | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_diff_self {f : α → β} {s : set α} {x : α} :
continuous_within_at f (s \ {x}) x ↔ continuous_within_at f s x | continuous_within_at_singleton.diff_iff | lemma | continuous_within_at_diff_self | topology | src/topology/continuous_on.lean | [
"topology.constructions"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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