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If $f$ is a continuous function from a set $S$ to a Euclidean space and $g$ is a linear function from a Euclidean space to a normed vector space, then the composition $g \circ f$ is continuous. |
If $h$ is a bilinear function and $f$ and $g$ are continuous functions, then the function $x \mapsto h(f(x), g(x))$ is continuous. |
If $h$ is bilinear and $f$ and $g$ are continuous, then $h(f(x), g(x))$ is continuous. |
If $f$ is a function from a set $X$ to a metric space $Y$ such that for all $x, y \in X$, we have $d(f(x), f(y)) \leq L \cdot d(x, y)$, where $L \geq 0$, then $f$ is $L$-Lipschitz on $X$. |
If $f$ is $L$-Lipschitz on $X$, then $f$ satisfies the Lipschitz condition on $X$. |
If $f$ is $L$-Lipschitz on $X$, then $L \geq 0$. |
If $f$ is Lipschitz on $X$, then $f$ is Lipschitz on $X$ with respect to the norm. |
If $f$ is $M$-Lipschitz on $E$ and $D \subseteq E$ and $M \leq L$, then $f$ is $L$-Lipschitz on $D$. |
If $f$ is Lipschitz on $S$, then $f$ is Lipschitz on any subset of $S$. |
If $f$ is a function from a linearly ordered set $X$ to a metric space $Y$ such that $f(x) \leq f(y)$ whenever $x \leq y$, then $f$ is Lipschitz continuous with Lipschitz constant $L$. |
Suppose $f$ and $g$ are L-Lipschitz functions on $[a,b]$ and $[b,c]$, respectively, and $f(b) = g(b)$. Then the function $h$ defined by $h(x) = f(x)$ if $x \leq b$ and $h(x) = g(x)$ if $x > b$ is L-Lipschitz on $[a,c]$. |
If $f$ is L-Lipschitz on $[a,b]$ and $g$ is M-Lipschitz on $[b,c]$, and $f(b) = g(b)$, then the function $h$ defined by $h(x) = f(x)$ if $x \leq b$ and $h(x) = g(x)$ if $x > b$ is $(\max\{L,M\})$-Lipschitz on $[a,c]$. |
If $f$ is L-Lipschitz on $X$, then $f$ is uniformly continuous on $X$. |
If $f$ is L-Lipschitz on $X$, then $f$ is continuous on $X$. |
If $f$ is L-Lipschitz on $X$, then $f$ is continuous at $x$ within $X$. |
If $f$ is differentiable on a convex set $X$ and the derivative of $f$ is bounded by $C$, then $f$ is $C$-Lipschitz on $X$. |
If $f$ is $C$-Lipschitz and $g$ is $D$-Lipschitz, then $g \circ f$ is $(D \cdot C)$-Lipschitz. |
If $f$ is $C$-Lipschitz and $g$ is $D$-Lipschitz, then $g \circ f$ is $(D \cdot C)$-Lipschitz. |
If two functions are equal on a set, then they are Lipschitz on that set if and only if they are Lipschitz on that set. |
If $f$ is a Lipschitz function on $U$, then so is $g$ if $g$ is equal to $f$ on $U$. |
A function is $C$-Lipschitz on the empty set if and only if $C \geq 0$. |
A function $f$ is $C$-Lipschitz on a set $X$ if and only if it is $C$-Lipschitz on $X \cup \{y\}$ for any $y \in X$. |
If $C \geq 0$, then $f$ is $C$-Lipschitz on the singleton set $\{x\}$ and on the empty set $\emptyset$. |
The identity function is 1-Lipschitz. |
The constant function $f(x) = c$ is $0$-Lipschitz on any set $U$. |
If $f$ and $g$ are Lipschitz functions, then $f + g$ is Lipschitz. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $|a| C$-Lipschitz on $U$. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $|a| C$-Lipschitz on $U$. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $(a C)$-Lipschitz on $U$ for any $a \geq 0$. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $(a C)$-Lipschitz on $U$ for any $a \geq 0$. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $(D C)$-Lipschitz on $U$, where $|a| \leq D$. |
If $f$ is $C$-Lipschitz on $U$, then $a f$ is $(D C)$-Lipschitz on $U$, where $|a| \leq D$. |
If $f$ is $C$-Lipschitz on $U$, then $-f$ is $C$-Lipschitz on $U$. |
The Lipschitz constant of a function $f$ is the same as the Lipschitz constant of $-f$. |
If $f$ and $g$ are Lipschitz functions on a set $U$, then $f - g$ is Lipschitz on $U$ with Lipschitz constant $C + D$. |
If $f$ is $C$-Lipschitz on $U$ and continuous on the closure of $U$, then $f$ is $C$-Lipschitz on the closure of $U$. |
If $f$ and $g$ are Lipschitz functions, then the function $(x, y) \mapsto (f(x), g(y))$ is Lipschitz. |
If $f$ is a $C$-Lipschitz function on $U$, then there exists a $C$-Lipschitz function $g$ on the closure of $U$ such that $g$ and $f$ agree on $U$. |
If $f$ is a bounded linear operator, then there exists a constant $B$ such that $f$ is $B$-Lipschitz on $A$. |
If $f$ is continuous on $[a,b]$ and locally Lipschitz on $(a,b)$, then $f$ is Lipschitz on $[a,b]$. |
If $f$ is continuous on $[a,b]$ and locally Lipschitz on $(a,b)$, then $f$ is Lipschitz on $[a,b]$. |
If $f$ is Lipschitz on each closed set $U_i$ and $\{u, v\} \subseteq \bigcup_{i \in I} U_i$, then $f$ is Lipschitz on $\{u, v\}$. |
If for every $t \in T$ and $x \in X$, there exists a neighborhood $U$ of $t$ and a neighborhood $V$ of $x$ such that $f$ is Lipschitz on $V$ uniformly for all $t \in U$, then $f$ is locally Lipschitz. |
Suppose $f$ is a local Lipschitz function from a topological space $X$ to a metric space $Y$. Then for every $x \in X$ and $t \in T$, there exists a neighborhood $U$ of $x$ and a neighborhood $V$ of $t$ such that $f$ is Lipschitz on $U \times V$. |
If $f$ is a local Lipschitz function, then for each $t \in T$, the function $f_t$ is continuous. |
If $f$ is locally Lipschitz on $g(T)$, then $f \circ g$ is locally Lipschitz on $T$. |
If $f(t,x)$ is continuous in $x$ for each $t$, then $f(t,x)$ is continuous in $(t,x)$. |
Suppose $f$ is a function from a compact set $T$ to the space of functions from a compact set $X$ to $\mathbb{R}$. If $f$ is locally Lipschitz, then it is Lipschitz. |
If $S \subseteq T$ and $Y \subseteq X$, then $f$ is locally Lipschitz on $S$ with respect to $Y$. |
The local Lipschitz constant of $-f$ is the same as the local Lipschitz constant of $f$. |
If $f$ and $g$ are locally Lipschitz functions, then so is the function $(f, g)$. |
If $f$ is a function from $S \times T$ to $\mathbb{R}$ such that for each $t \in T$, the function $f(t, \cdot)$ is Lipschitz continuous on $S$, then $f$ is locally Lipschitz continuous on $S \times T$. |
If $f$ is a bounded linear operator, then it is locally Lipschitz. |
If $f$ is continuously differentiable, then it is locally Lipschitz. |
If $S$ is a finite set and for each $i \in S$, the set $\{x \in \Omega \mid P_i(x)\}$ is in $M$, then the set $\{x \in \Omega \mid \forall i \in S, P_i(x)\}$ is in $M$. |
The predicate $P$ is measurable if and only if the set $\{x \in X : P(x)\}$ is measurable. |
If $P$ is a measurable predicate on $M$, and $f$ is a measurable function from $N$ to $M$, then $P \circ f$ is a measurable predicate on $N$. |
If $A$ is a measurable set in $N$ and $f$ is a measurable function from $M$ to $N$, then the preimage of $A$ under $f$ is a measurable set in $M$. |
If $P$ is a predicate on a measurable space $M$, then the set $\{x \in M \mid P(x)\}$ is measurable. |
If $K$ is a proposition and $P$ is a predicate, then the proposition $K \rightarrow P$ is a predicate. |
If $K$ is a measurable predicate and $P$ is a measurable predicate, then the predicate $K \wedge P$ is measurable. |
If $K$ is true and $P$ is a measurable predicate, then $P \wedge K$ is a measurable predicate. |
If $P$ is a predicate on a measurable space $(X, \mathcal{A})$, then the predicate $K \vee P$ is also measurable. |
If $P$ is a predicate on $M$ and $K$ is a proposition, then $P \vee K$ is a predicate on $M$. |
A predicate is a measurable function from a measurable space to the space of truth values. |
If $P(x,i)$ is a predicate on $x$ that depends on a countable index $i$, then the predicates $\forall i. P(x,i)$ and $\exists i. P(x,i)$ are measurable. |
If $X$ is a countable set and $N$ is a function from $X$ to the set of subsets of $M$, then the intersection and union of the sets $N(x)$ for $x \in X$ are measurable. |
If $I$ is a finite set and $P$ is a predicate on $M$, then the following predicates are also predicates on $M$: $\{x \in M \mid x \in \bigcap_{i \in I} N(x, i)\}$ $\{x \in M \mid x \in \bigcup_{i \in I} N(x, i)\}$ $\{x \in M \mid \forall i \in I, P(x, i)\}$ $\{x \in M \mid \exists i \in I, P(x, i)\}$ |
If $I$ is a countable set and $N_i \in \mathcal{M}$ for all $i \in I$, then $\bigcup_{i \in I} N_i \in \mathcal{M}$ and $\bigcap_{i \in I} N_i \in \mathcal{M}$. |
If $A$ and $B$ are measurable sets, then $A \cap B$ is a measurable set. |
If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, and $c$ is a constant in $N$, then the predicate $f(x) = c$ is measurable. |
If $f$ is a measurable function from a measurable space $M$ to the set of all natural numbers, then the predicate $\{x \in M \mid f(x) = c\}$ is measurable. |
If $f$ is a measurable function from a measurable space $M$ to the set of natural numbers, then the predicate $\{x \in M \mid f(x) = c\}$ is measurable. |
If $f$ is a measurable function from $M$ to $N$ and $c$ is a constant in $N$, then the predicate $\{x \in M \mid f(x) \leq c\}$ is measurable. |
If $f$ is a measurable function from a measurable space $M$ to a measurable space $N$, and $c$ is a constant in $N$, then the set $\{x \in M \mid c \leq f(x)\}$ is measurable. |
If $f$ is a measurable function from $M$ to $N$ and $c$ is a constant such that the set $\{x \in N \mid x < c\}$ is measurable, then the set $\{x \in M \mid f(x) < c\}$ is measurable. |
If $f$ is a measurable function from $M$ to $N$ and $c$ is a constant in $N$, then the predicate $\{x \in M \mid c < f(x)\}$ is measurable. |
If $P$ is a predicate, and $A$ and $B$ are measurable functions, then the function $x \mapsto \mathbf{1}_{P}(x) A(x) + \mathbf{1}_{\lnot P}(x) B(x)$ is measurable. |
If $A$ is a function from $I$ to the measurable sets of $M$, then $A(i)$ is measurable for all $i \in I$. |
If $A$ is a set of sets of $N$ and $f$ is a measurable function from $M$ to $N$, then the set $\{x \in M \mid f(x) \in A\}$ is measurable. |
If $A_i$ is a measurable set for each $i$, then $A_i$ is a measurable set for each $i$. |
If $A_i$ is a measurable set in the $i$th $\sigma$-algebra, and $f$ is a measurable function from $M$ to $N_i$, then the set $\{x \in M \mid f(x) \in A_i\}$ is measurable in $M$. |
If $A_i$ is a measurable set for each $i \in I$, then $A_i$ is a measurable set for each $i \in I$. |
If $f$ is measurable from $M$ to $N_i$, then the predicate $\{x \in M \mid f(x) \in A_i\}$ is measurable. |
If $S$ is a measurable function from a measurable space $M$ to the set of subsets of the natural numbers, then the set of points $x$ in $M$ such that $S(x)$ is finite is measurable. |
If $P_i$ is a sequence of measurable functions, then the function $x \mapsto \min\{i \mid P_i(x)\}$ is measurable. |
If $P_i$ is a measurable predicate for each $i$, then the function $x \mapsto \max\{i \mid P_i(x)\}$ is measurable. |
If $P_i$ is a measurable predicate for each $i$, then the function $x \mapsto \min\{i \mid P_i(x)\}$ is measurable. |
If $s$ is an element of $S$ and $A$ is a subset of $S$, then $A \cup \{s\}$ is a subset of $S$. |
Every set is measurable. |
If $S$ is a measurable function from a measurable space $M$ to the set of subsets of the natural numbers, then the function $x \mapsto \#(S(x))$ is measurable. |
If $X$ is a countable set and $P$ is a predicate on $X$, then the predicate $\forall i \in X. P(i)$ is measurable. |
The empty set is measurable. |
The constant function $f(x) = 1$ is measurable. |
If $F$ is a countable family of measurable functions, then the pointwise supremum of $F$ is measurable. |
If $F_i$ is a measurable function for each $i \in I$, where $I$ is countable, then $\inf_{i \in I} F_i$ is measurable. |
If $F$ is a sup-continuous function from the set of measurable functions to itself, and if $P$ is a property of measurable functions such that $P(M)$ implies that $F(A)$ is measurable whenever $A$ is measurable, then the least fixed point of $F$ is measurable. |
If $F$ is a sup-continuous operator on the space of measurable functions from $M$ to $\mathbb{N}$, then the least fixed point of $F$ is measurable. |
If $F$ is an inf-continuous operator on the space of measurable functions, and if $P$ is a property of measurable functions such that $P(M)$ implies that $A \in \mathcal{M}(N)$ for all $N$ such that $P(N)$ holds, then $F(A) \in \mathcal{M}(M)$ for all $M$ such that $P(M)$ holds, then the greatest fixed point of $F$ is ... |
If $F$ is an inf-continuous function from the set of measurable functions to itself, then the greatest fixed point of $F$ is measurable. |
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