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Suppose $f$ and $g$ are functions from $\mathbb{R}$ to a topological space $X$, and $f$ converges to $l$ at $a$. If $g$ and $f$ agree on a neighborhood of $a$, then $g$ converges to $l$ at $a$. |
If $f$ and $g$ are functions such that $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$. |
If $f$ converges to $0$ and $g$ is bounded above by $f$ and bounded below by $0$, then $g$ converges to $0$. |
A function $f$ is continuous at $a$ if and only if the function $h \mapsto f(a + h)$ is continuous at $0$. |
A function $f$ is continuous at $x$ if and only if the function $h \mapsto f(x + h)$ converges to $f(x)$ as $h$ approaches $0$. |
If $f$ is continuous at $a$ and $g$ is continuous at $f(a)$, then $g \circ f$ is continuous at $a$. |
If $f$ is continuous at $a$, then $\|f\|$ is continuous at $a$. |
If $f$ is continuous at $a$, then $\lvert f \rvert$ is continuous at $a$. |
If $f$ and $g$ are continuous at $a$, then $f + g$ is continuous at $a$. |
If $f$ is continuous at $a$, then $-f$ is continuous at $a$. |
If $f$ and $g$ are continuous at $a$, then $f - g$ is continuous at $a$. |
If $f$ and $g$ are continuous at $a$, then $f \cdot g$ is continuous at $a$. |
If $g$ is a continuous function, then $f \circ g$ is continuous. |
If $f$ and $g$ are continuous at $a$, then the function $f \cdot g$ is continuous at $a$. |
The function $f(x) = ax$ is continuous. |
The function $x \mapsto x$ is continuous. |
If $f$ is continuous at $a$, then $f^n$ is continuous at $a$. |
If $f_i$ is continuous at $a$ for all $i \in A$, then $\sum_{i \in A} f_i$ is continuous at $a$. |
A function $f$ is uniformly continuous on a set $S$ if and only if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x, x' \in S$, if $|x - x'| < \delta$, then $|f(x) - f(x')| < \epsilon$. |
A function $f$ is uniformly continuous if and only if for every $\epsilon > 0$, there exists $\delta > 0$ such that for all $x, y \in \mathbb{R}$, if $|x - y| < \delta$, then $|f(x) - f(y)| < \epsilon$. |
If $f$ is uniformly continuous, then $f$ is continuous. |
If $f$ is uniformly continuous on a set $S$, and if the sequence $X$ is Cauchy and takes values in $S$, then the sequence $f(X)$ is Cauchy. |
If $f$ is uniformly continuous and $X$ is a Cauchy sequence, then $f(X)$ is a Cauchy sequence. |
If $f$ is uniformly continuous on $S$ and $\sigma$ is a Cauchy sequence in $S$, then $f \circ \sigma$ is a Cauchy sequence. |
If $f$ is a bounded linear operator, then $f$ is uniformly continuous. |
If $X$ is a Cauchy sequence, then $f(X)$ is a Cauchy sequence. |
If $f$ is continuous at $x$ and $f$ is non-negative on the interval $(b, x)$, then $f(x) \geq 0$. |
If $f$ and $g$ are two nested sequences of real numbers such that $f_n \leq g_n$ for all $n$ and $\lim_{n \to \infty} (f_n - g_n) = 0$, then there exists a real number $l$ such that $\lim_{n \to \infty} f_n = l = \lim_{n \to \infty} g_n$. |
If $P$ is a transitive relation on the real numbers and $P$ is locally true, then $P$ is true. |
The closed interval $[a, b]$ is compact. |
If $f$ is a continuous function on the closed interval $[a,b]$, then $f([a,b])$ is also a closed interval. |
If $f$ is a continuous real-valued function defined on a set $s$, then there exists an open set $A$ such that $A \cap s = \{x \in s \mid f(x) > 0\}$. |
If $f$ and $g$ are continuous real-valued functions defined on a set $S$, then the set $\{x \in S \mid f(x) < g(x)\}$ is open in $S$. |
If $f$ is continuous on the closed interval $[a,b]$, then $f$ attains its supremum on $[a,b]$. |
If $f$ is continuous on the closed interval $[a,b]$, then $f$ attains its least upper bound on $[a,b]$. |
If $f$ is continuous on the closed interval $[a,b]$, then $f$ is bounded on $[a,b]$. |
If $f$ is continuous on the closed interval $[a,b]$, then $f$ has a supremum on $[a,b]$. |
If $f$ is continuous on the closed interval $[a,b]$, then $f$ attains its maximum and minimum on $[a,b]$. |
Suppose $f$ is a function defined on an interval $(x-d, x+d)$ and $g$ is its inverse. If $f$ is continuous at $x$, then $g$ is continuous at $f(x)$. |
Suppose $f$ and $g$ are functions such that $g(f(x)) = x$ for all $x$ in the interval $(a,b)$. If $f$ is continuous at $x$, then $g$ is continuous at $f(x)$. |
If $f$ converges to a positive number $l$, then there exists a positive number $r$ such that $f$ is positive on the interval $(c - r, c + r)$. |
If $f$ converges to $l < 0$, then there exists $r > 0$ such that for all $x$ with $|x - c| < r$, we have $f(x) < 0$. |
If $f$ converges to $l \neq 0$, then there exists $r > 0$ such that $f(x) \neq 0$ for all $x$ with $|x - c| < r$. |
If $f$ is a linear operator, then $f$ satisfies the usual linear properties. |
The set of values of a function $f$ on a finite set is finite. |
If $d$ is a subset of the basis of a Euclidean space, then the expansion of a vector $x$ in terms of the basis vectors in $d$ is unique. |
If $d$ is a subset of the standard basis, then $d$ is independent. |
For any real vector space $V$ and any $a \in V$, the set $a + S$ is a subset of $a + T$ if and only if $S$ is a subset of $T$. |
The translation $x \mapsto a + x$ is injective on any set $A$. |
For any set $S$ and any elements $a$ and $b$ of an abelian group, the set of all elements of the form $b + (a + x)$ for $x \in S$ is equal to the set of all elements of the form $(a + b) + x$ for $x \in S$. |
If two sets are equal after translation by the same amount, then they are equal. |
For any set $S$ and any element $a$, the set $T = \{a + x \mid x \in S\}$ is equal to the set $\{-a + x \mid x \in S\}$. |
If $V$ is a subset of the translation of $S$ by $-a$, then $V$ is a subset of the translation of $S$ by $a$. |
If $f$ is a linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$, then the $j$th component of $f(x)$ is equal to the sum of the $j$th components of $f(e_i)$ times the $i$th component of $x$, where $e_i$ is the $i$th standard basis vector. |
Two vectors are equal if and only if their inner products with themselves are equal and their inner products with each other are equal. |
If $|y - x_1| < \frac{\epsilon}{2}$ and $|y - x_2| < \frac{\epsilon}{2}$, then $|x_1 - x_2| < \epsilon$. |
If $|x - y| < \frac{e}{2}$ and $|x' - y| < \frac{e}{2}$, then $|x - x'| < e$. |
If $|y - x_1| < \frac{e}{2}$ and $|y - x_2| < \frac{e}{2}$, then $|x_1 - x_2| < e$. |
If $|x - y| < \frac{\epsilon}{2}$ and $|x' - y| < \frac{\epsilon}{2}$, then $|x - x'| < \epsilon$. |
The sum of the empty set is zero. If $S$ is a finite set and $x \in S$, then the sum of $S$ is the same as the sum of $S \setminus \{x\}$. If $S$ is a finite set and $x \notin S$, then the sum of $S$ is the same as the sum of $S \cup \{x\}$. |
Two vectors are equal if and only if they have the same dot product with every vector. |
Two vectors are equal if and only if they are equal when dotted with any other vector. |
If $n$ is less than or equal to the cardinality of $A$, then there exists a subset $S$ of $A$ such that $S$ has cardinality $n$. |
The set of all vectors in $\mathbb{R}^n$ whose $i$th coordinate is zero whenever $P(i)$ is true is a subspace. |
The dimension of the set of points in $\mathbb{R}^n$ that are orthogonal to a given set of $d$ linearly independent vectors is $d$. |
Two vectors are orthogonal if and only if they are equal to zero. |
The following are equivalent: $a$ is orthogonal to $0$. $a$ is orthogonal to $x$ and $c$ is a scalar. $a$ is orthogonal to $x$ and $-x$. $a$ is orthogonal to $x$ and $y$ and $x + y$. $a$ is orthogonal to $x$ and $y$ and $x - y$. $0$ is orthogonal to $a$. $x$ is orthogonal to $a$ and $c$ is a scalar. $x$ is orthogonal t... |
Two vectors are orthogonal if and only if they are orthogonal. |
If $c \neq 0$, then $cx$ is orthogonal to $y$ if and only if $x$ is orthogonal to $y$. |
If $f_i$ and $g_j$ are pairwise orthogonal, then so are $a_i f_i$ and $a_j g_j$. |
If $x$ is orthogonal to each $f(y)$ for $y \in s$, then $x$ is orthogonal to $\sum_{y \in s} f(y)$. |
If $f$ is a finite set of vectors and $y$ is a vector such that $f(x)$ is orthogonal to $y$ for all $x \in f$, then the sum of the vectors in $f$ is orthogonal to $y$. |
If two vectors are orthogonal, then the Pythagorean theorem holds for them. |
If $f_i$ are pairwise orthogonal, then $\|\sum_i f_i\|^2 = \sum_i \|f_i\|^2$. |
A linear transformation $f$ is an orthogonal transformation if and only if $\|f(v)\| = \|v\|$ for all $v$. |
The identity function is an orthogonal transformation. |
If $f$ is an orthogonal transformation, then $f(x)$ is orthogonal to $f(y)$ if and only if $x$ is orthogonal to $y$. |
If $f$ and $g$ are orthogonal transformations, then so is $f \circ g$. |
A linear transformation $f$ is orthogonal if and only if $-f$ is orthogonal. |
If $f$ is an orthogonal transformation, then $f(c \cdot v) = c \cdot f(v)$. |
If $f$ is an orthogonal transformation, then $f$ is linear. |
If $f$ is an orthogonal transformation, then $f$ is injective. |
If $f$ is an orthogonal transformation, then $f$ is surjective. |
If $f$ is an orthogonal transformation, then $f$ is a bijection. |
If $f$ is an orthogonal transformation, then $f^{-1}$ is an orthogonal transformation. |
If $f$ is an orthogonal transformation, then $\|f(x)\| = \|x\|$. |
If $h$ is bilinear, then $h(x + y, z) = h(x, z) + h(y, z)$. |
If $h$ is a bilinear function, then $h(x, y + z) = h(x, y) + h(x, z)$. |
The multiplication operation is bilinear. |
If $h$ is a bilinear function, then $h(cx, y) = ch(x, y)$. |
If $h$ is bilinear, then $h(x, c y) = c h(x, y)$. |
If $h$ is bilinear, then $h(-x)y = -h(x)y$. |
If $h$ is bilinear, then $h(x, -y) = -h(x, y)$. |
In an abelian group, $x = x + y$ if and only if $y = 0$. |
If $h$ is a bilinear function, then $h(0, x) = 0$ for all $x$. |
If $h$ is bilinear, then $h(x, 0) = 0$. |
If $h$ is bilinear, then $h(x - y)z = h(x)z - h(y)z$. |
If $h$ is bilinear, then $h(z, x - y) = h(z, x) - h(z, y)$. |
If $h$ is bilinear, then $h( \sum_{i \in S} f_i, \sum_{j \in T} g_j) = \sum_{(i,j) \in S \times T} h(f_i, g_j)$. |
If $f$ is an adjoint of $g$, then $g$ is an adjoint of $f$. |
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