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If $f$ and $g$ are continuous functions, then so is $f \cdot g$. |
If $f$ and $g$ are continuous functions, then so is $f^g$. |
If $f$ and $g$ are continuous functions on a set $A$, then the function $h(x) = f(x)g(x)$ is continuous on $A$. |
If $f$ and $g$ are continuous functions from $A$ to a topological monoid, then the function $x \mapsto f(x)^{g(x)}$ is continuous. |
If $f$ and $g$ converge to $1$, then $f \cdot g$ converges to $1$. |
If $f_i$ converges to $a_i$ for each $i \in I$, then $\prod_{i \in I} f_i$ converges to $\prod_{i \in I} a_i$. |
If $f_i(x) \to 1$ for each $i$, then $\prod_i f_i(x) \to 1$. |
If $f_i$ is continuous for each $i \in I$, then the product $\prod_{i \in I} f_i$ is continuous. |
If $f_i$ is continuous for each $i \in I$, then the product $\prod_{i \in I} f_i$ is continuous. |
If $f$ is a zero-preserving function and $g$ is a bounded function, then the product $f \cdot g$ is a zero-preserving function. |
If $f$ is a bounded bilinear function and $g$ is a zero function, then the product $f \cdot g$ is a zero function. |
If $f$ is a function that converges to a nonzero number $a$, then the function $1/f$ is bounded. |
If $f$ converges to $a$ and $a \neq 0$, then $1/f$ converges to $1/a$. |
If $f$ is a continuous function from a topological space $X$ to a normed algebra $A$ and $f(x) \neq 0$ for some $x \in X$, then the function $g(x) = f(x)^{-1}$ is continuous. |
If $f$ is a continuous function on a set $S$ and $f(a) \neq 0$, then the function $1/f$ is continuous at $a$. |
If $f$ is a continuous function from a set $S$ to a real normed division algebra, and $f(x) \neq 0$ for all $x \in S$, then the function $x \mapsto \frac{1}{f(x)}$ is continuous on $S$. |
If $f$ and $g$ are functions that tend to $a$ and $b$ respectively, and $b \neq 0$, then the function $f/g$ tends to $a/b$. |
If $f$ and $g$ are continuous functions on a topological space $F$ and $g$ is nonzero at the limit of $F$, then the function $f/g$ is continuous on $F$. |
If $f$ and $g$ are continuous functions on a set $S$, and $g(a) \neq 0$, then the function $f/g$ is continuous at $a$. |
If $f$ and $g$ are continuous functions at $a$ and $g(a) \neq 0$, then the function $f/g$ is continuous at $a$. |
If $f$ and $g$ are continuous functions on a set $S$ and $g(x) \neq 0$ for all $x \in S$, then the function $f/g$ is continuous on $S$. |
If $f$ tends to $a$ and $a \neq 0$, then $f^n$ tends to $a^n$. |
If $f$ is a continuous function from a topological space $X$ to a real normed algebra $A$ and $f(x) \neq 0$ for some $x \in X$, then the function $x \mapsto f(x)^n$ is continuous. |
If $f$ is a continuous function on a set $S$ and $f(a) \neq 0$, then the function $x \mapsto f(x)^n$ is continuous at $a$ on $S$. |
If $f$ is a continuous function from $s$ to a real normed division algebra, and $f(x) \neq 0$ for all $x \in s$, then the function $x \mapsto f(x)^n$ is continuous on $s$. |
If $f$ converges to $l$ and $l \neq 0$, then $\text{sgn}(f)$ converges to $\text{sgn}(l)$. |
If $f$ is a continuous function from a topological space $X$ to a normed vector space $Y$ and $f(x) \neq 0$ for some $x \in X$, then the function $g(x) = \text{sgn}(f(x))$ is continuous. |
If $f$ is a continuous function defined on a set $S$ and $f(a) \neq 0$, then the sign function of $f$ is continuous at $a$. |
If $f$ is a continuous function from a topological space to a normed vector space, and $f(a) \neq 0$, then the sign function of $f$ is continuous at $a$. |
If $f$ is a continuous function from a set $S$ to $\mathbb{R}^n$ such that $f(x) \neq 0$ for all $x \in S$, then the function $g(x) = \frac{f(x)}{\|f(x)\|}$ is continuous on $S$. |
If $f$ is a nonnegative real-valued function, then $f$ tends to infinity if and only if for every $r > c$, there exists an $x$ such that $r \leq f(x)$. |
If $f$ tends to infinity, then $\|f\|$ tends to infinity. |
If the norm of a function $f$ tends to infinity, then $f$ tends to infinity. |
The norm of a real number tends to infinity as the real number tends to infinity. |
The limit of $f$ at infinity is the same as the limit of $f$ at infinity. |
For any real-valued vector $a$, there exists a real number $M$ such that for all $x > M$, we have $x \neq a$. |
If $f(n)$ converges to $L$ as $n \to \infty$, then $f(n)$ converges to $L$ as $n \to \infty$. |
The sequence of integers tends to infinity. |
The sequence of integers tends to infinity. |
The absolute value function $|\cdot|$ is a limit of the real numbers. |
The function $f(x) = x$ is a limit of the function $f(x) = x^2$ as $x$ approaches infinity. |
If $f$ converges to $c$ and diverges to infinity, then $F$ is not a filter. |
If $f$ tends to infinity, then $f$ is not convergent. |
If $f$ tends to infinity, then eventually $f$ is not equal to $c$. |
The sequence of natural numbers tends to infinity. |
The filterlim function can be used to split a real-valued function at a real number. |
The filter of neighborhoods of $a$ is the same as the filter of neighborhoods of $a - d$. |
The filter of neighborhoods of $-a$ is the same as the filter of neighborhoods of $a$ under the map $x \mapsto -x$. |
The filter of neighbourhoods of $a$ is mapped to the filter of neighbourhoods of $a - d$ by the map $x \mapsto x - d$. |
The filter map $\lambda x \mapsto x - d$ applied to the filter at right $a$ is the filter at right $a - d$. |
The filter at_right a is equal to the filter at_right 0 shifted by a. |
The limit of $f(x)$ as $x$ approaches $a$ from the right is the same as the limit of $f(x + a)$ as $x$ approaches $0$ from the right. |
For any real number $a$, the following are equivalent: $\forall \epsilon > 0, \exists x > a, P(x)$ $\forall \epsilon > 0, \exists x > 0, P(x + a)$ |
The filter at $a$ is the same as the filter at $0$ after translation by $a$. |
The limit of $f(x)$ as $x$ approaches $a$ is the same as the limit of $f(x + a)$ as $x$ approaches $0$. |
For any real-valued function $f$ defined on a neighborhood of $a$, the following are equivalent: $f$ is eventually positive in a neighborhood of $a$. $f(x + a)$ is eventually positive in a neighborhood of $0$. |
The filter map of the negation function at a point $a$ is the same as the filter at $-a$. |
The filter of points at the left of $a$ is the image of the filter of points at the right of $-a$ under the map $x \mapsto -x$. |
The filter at_right a is equal to the filter at_left (-a) after applying the map $x \mapsto -x$. |
The limit of $f$ at $a$ from the left is the same as the limit of $f(-x)$ at $-a$ from the right. |
For any real number $a$, the following are equivalent: $P$ holds for all $x < a$ sufficiently close to $a$. $P$ holds for all $x > -a$ sufficiently close to $-a$. |
The limit of $-x$ as $x$ approaches $-\infty$ is $\infty$. |
The limit of $-x$ as $x$ approaches infinity is $-\infty$. |
The filter at the bottom of a linearly ordered abelian group is the mirror image of the filter at the top. |
The filter at_top is the mirror image of the filter at_bot. |
The filter $\{f(x) \mid x \geq a\}$ is the same as the filter $\{f(-x) \mid x \leq -a\}$. |
The filter limit of $f$ at $-\infty$ is the same as the filter limit of $f$ at $\infty$. |
The filter $\{f(x) \mid x \in F\}$ converges to $\infty$ if and only if the filter $\{-f(x) \mid x \in F\}$ converges to $-\infty$. |
If $f$ converges to $y$ along any sequence that converges to $-\infty$, then $f$ converges to $y$ at $-\infty$. |
If $f$ tends to infinity along a filter $F$ and $f$ is eventually positive along $F$, then $f$ tends to infinity along $F$. |
If $f$ tends to infinity along a filter $F$, and $f$ is eventually negative along $F$, then $f$ tends to $-\infty$ along $F$. |
The filter $\{f \leq a\}$ converges to $-\infty$ if and only if the filter $\{-f \geq -a\}$ converges to $\infty$. |
The limit of the inverse of $x$ as $x$ approaches $0$ from the right is infinity. |
The inverse function tends to zero at infinity. |
If $f$ converges to $c$ and $g$ tends to infinity, then $f + g$ tends to infinity. |
If $f$ tends to infinity and $g$ converges, then $f + g$ tends to infinity. |
The limit of the sequence $1/x$ as $x$ approaches infinity is $0$ from the right. |
If $f$ is a function that converges to $0$ and is eventually positive, then the inverse of $f$ tends to infinity. |
The limit of the inverse of $x$ as $x$ approaches $0$ from the left is $-\infty$. |
If $f$ converges to $0$ and is eventually negative, then $1/f$ converges to $-\infty$. |
The filter of right neighborhoods of $0$ is equal to the filter of neighborhoods of $\infty$ under the map $x \mapsto 1/x$. |
For any property $P$, the following are equivalent: $P$ holds for all $x > 0$ sufficiently close to $0$. $P$ holds for all $x > 0$ sufficiently large. |
The limit of $f$ at the right of $0$ is the same as the limit of $f(1/x)$ as $x$ goes to infinity. |
The filter of events that happen at infinity is the same as the filter of events that happen to the right of $0$. |
The statement $P(x)$ holds eventually as $x \to \infty$ if and only if the statement $P(1/x)$ holds eventually as $x \to 0^+$. |
The function $f$ tends to infinity if and only if the function $1/f$ tends to zero from the right. |
The function $f(x) = 1/x$ tends to infinity as $x$ tends to $0$. |
The limit of the inverse of a function $g$ at $0$ exists if and only if the limit of $g$ at $\infty$ exists. |
If $f$ tends to $c$ and $g$ tends to infinity, then $f \cdot g$ tends to infinity. |
If $f(x)$ tends to infinity, then $1/f(x)$ tends to zero. |
If $f$ and $g$ are real-valued functions such that $f$ converges to $c$ and $g$ tends to infinity, then $f/g$ tends to zero. |
If $c$ is a positive integer, then the sequence $c, 2c, 3c, \ldots$ converges to infinity. |
If $c$ is a positive integer, then the sequence $x_n = nc$ tends to infinity. |
If $f$ converges to $p$ from the right, then $c f$ converges to $c p$ from the right, for any $c > 0$. |
If $c \neq 0$, then the filter map $x \mapsto cx$ is a homeomorphism from the neighborhood filter of $a$ to the neighborhood filter of $ca$. |
If $c > 0$, then the filter map $f(x) = cx$ maps the filter at $p$ to the filter at $cp$. |
The filter of neighborhoods of $0$ is equal to the filter of neighborhoods of infinity. |
If $f$ converges to $l$ at infinity, then $f(1/x)$ converges to $l$ at $0$. |
If the limit of $f(1/x)$ as $x$ approaches $0$ is $l$, then the limit of $f(x)$ as $x$ approaches $\infty$ is $l$. |
If $f$ tends to a positive number $c$ and $g$ tends to infinity, then $f \cdot g$ tends to infinity. |
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