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If $f$ is a Z-function and $r > 0$, then $f$ is eventually less than $r$. |
If $f$ and $g$ are two functions such that $f(x) = g(x)$ for all $x$ in some set $F$, then $f$ is a zero function on $F$ if and only if $g$ is a zero function on $F$. |
The zero function is a Z-function. |
The zero set of a function $f$ is the same as the zero set of the function $x \mapsto \|f(x)\|$. |
If $f$ is a zero function and $g$ is bounded by $Kf$, then $g$ is a zero function. |
If $f$ is a Z-function and $g$ is a Z-function such that $|f(x)| \leq |g(x)|$ for all $x$, then $f$ is a Z-function. |
If $f$ and $g$ are functions from a set $X$ to a field $F$ such that $f(x) + g(x)$ is defined for all $x \in X$, then $f + g$ is a function from $X$ to $F$. |
If $f$ is a zero function on a field $F$, then $-f$ is also a zero function on $F$. |
If $f$ and $g$ are zero-free functions, then so is $f - g$. |
If $g$ is a zero function, then $f \circ g$ is a zero function. |
If $f$ and $g$ are zero-sum free, then so is $f \cdot g$. |
If $f$ is a zero-preserving function from a vector space $V$ to a vector space $W$, then the function $x \mapsto f(x) + a$ is also zero-preserving. |
If $f$ is a zero function, then $a \cdot f$ is also a zero function. |
The function $f(x,y) = xy$ is bounded bilinear. |
If $f$ is a bounded bilinear function, then $f(x, \cdot)$ is a bounded linear function. |
The function $f(x, y) = xy$ is bounded bilinear. |
$f \to a$ if and only if $f - a \to 0$. |
If $f$ converges to $0$ and $g$ is eventually bounded by $Kf$, then $g$ converges to $0$. |
If $f$ and $g$ converge to $l$ and $m$, respectively, then $f - g$ converges to $l - m$. |
If $f$ and $g$ are continuous functions, then the function $x \mapsto \text{dist}(f(x), g(x))$ is continuous. |
If $f$ and $g$ are continuous functions from $S$ to a metric space, then the function $x \mapsto \text{dist}(f(x), g(x))$ is continuous. |
The function $x \mapsto \|x - a\|$ is continuous at $b$. |
If $f$ converges to $a$, then $\|f\|$ converges to $\|a\|$. |
If $f$ is a continuous function, then the function $x \mapsto \|f(x)\|$ is continuous. |
If $f$ is continuous on $S$, then the function $x \mapsto \|f(x)\|$ is continuous on $S$. |
The norm function is continuous on any set $S$. |
If $f$ converges to $0$, then $\|f\|$ converges to $0$. |
If the norm of a function tends to zero, then the function itself tends to zero. |
The sequence $f$ converges to $0$ if and only if the sequence $|f|$ converges to $0$. |
If $f$ converges to $l$, then $\lvert f \rvert$ converges to $\lvert l \rvert$. |
If $f$ is a continuous function from a topological space $X$ to the real numbers, then the function $x \mapsto |f(x)|$ is continuous. |
If $f$ is continuous on $S$, then $\lvert f \rvert$ is continuous on $S$. |
If $f$ converges to $0$, then $\lvert f \rvert$ converges to $0$. |
If the absolute value of a function tends to zero, then the function itself tends to zero. |
The sequence $f$ converges to $0$ if and only if the sequence $\lvert f \rvert$ converges to $0$. |
If $f$ and $g$ converge to $a$ and $b$, respectively, then $f + g$ converges to $a + b$. |
If $f$ and $g$ are continuous functions, then so is $f + g$. |
If $f$ and $g$ are continuous functions on a set $S$, then the function $f + g$ is continuous on $S$. |
If $f$ and $g$ converge to $0$ in the filter $F$, then $f + g$ converges to $0$ in $F$. |
If $f_i$ converges to $a_i$ for each $i \in I$, then $\sum_{i \in I} f_i$ converges to $\sum_{i \in I} a_i$. |
If $f_i(x) \to 0$ for all $i$, then $\sum_i f_i(x) \to 0$. |
If $f_i$ is continuous for each $i \in I$, then $\sum_{i \in I} f_i$ is continuous. |
If $f_i$ is continuous for each $i \in I$, then $\sum_{i \in I} f_i$ is continuous. |
If $f$ tends to $a$ in $F$, then $-f$ tends to $-a$ in $F$. |
If $f$ is continuous, then so is $-f$. |
If $f$ is continuous on $s$, then $-f$ is continuous on $s$. |
If $f$ tends to $a$, then $-f$ tends to $-a$. |
If $f$ tends to $-y$ in $F$, then $-f$ tends to $y$ in $F$. |
If $f$ and $g$ converge to $a$ and $b$, respectively, then $f - g$ converges to $a - b$. |
If $f$ and $g$ are continuous functions, then so is $f - g$. |
If $f$ and $g$ are continuous functions on a set $S$, then so is $f - g$. |
The function $f(y) = y - x$ is continuous on any set $s$. |
If $f$ and $g$ are real-valued functions and $f(x) \leq g(x)$ for all $x$ in some set $S$, and if $f(x)$ and $g(x)$ both tend to $L$ as $x$ tends to $a$, then $L$ is the limit of $f(x)$ and $g(x)$ as $x$ tends to $a$. |
The function $x \mapsto c x$ is linear. |
If $g$ tends to $a$ in $F$, then $f \circ g$ tends to $f(a)$ in $F$. |
If $f$ is a bounded linear operator and $g$ is a continuous function, then $f \circ g$ is continuous. |
If $f$ is a bounded linear operator and $g$ is a continuous function, then $f \circ g$ is continuous. |
If $g$ tends to $0$ in $F$, then $f \circ g$ tends to $0$ in $F$. |
If $f$ and $g$ are bounded bilinear functions, then $f \cdot g$ is also bounded bilinear. |
If $f$ and $g$ are continuous functions from a topological space $X$ to a normed vector space $V$, then the function $h(x) = f(x) \cdot g(x)$ is continuous. |
If $f$ and $g$ are continuous functions from a set $S$ to a normed vector space $V$, then the function $h$ defined by $h(x) = f(x) \cdot g(x)$ is continuous. |
If $f$ and $g$ converge to $0$ in a topological vector space, then $f \cdot g$ converges to $0$. |
If $f$ tends to $0$ in a filter $F$, then $f \cdot c$ tends to $0$ in $F$. |
If $f$ tends to $0$ in a filter $F$, then $c f$ tends to $0$ in $F$. |
If $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$, then $f$ is continuous if and only if $\lim_{x \to a} f(x) = f(a)$ for all $a \in \mathbb{R}$. |
If $f$ tends to $a$ and $g$ tends to $b$, then $f \cdot g$ tends to $a \cdot b$. |
If $f$ and $g$ converge to $a$ and $b$, respectively, then $f \cdot g$ converges to $a \cdot b$. |
If $f$ converges to $l$ in $F$, then $c \cdot f$ converges to $c \cdot l$ in $F$. |
If $f$ converges to $l$ in $F$, then $f \cdot c$ converges to $l \cdot c$ in $F$. |
If $c$ is a nonzero real number, then the limit of $c \cdot f(x)$ as $x$ approaches $c \cdot l$ is $F$ if and only if the limit of $f(x)$ as $x$ approaches $l$ is $F$. |
If $c \neq 0$, then $\lim_{x \to l} f(x) c = \lim_{x \to l} f(x)$. |
If $c$ is a nonzero constant, then the sequence $c \cdot a_n$ converges to $0$ if and only if the sequence $a_n$ converges to $0$. |
If $c \neq 0$, then $\lim_{n \to \infty} a_n c = 0$ if and only if $\lim_{n \to \infty} a_n = 0$. |
If $c \neq 0$, then $\lim_{n \to \infty} \frac{a_n}{c} = 0$ if and only if $\lim_{n \to \infty} a_n = 0$. |
The sequence $a/n$ converges to $0$ for any $a \in \mathbb{R}$. |
The function $x \mapsto x$ is continuous. |
The function $f(x, y) = xy$ is continuous. |
The multiplication function is continuous. |
The function $x \mapsto \mathrm{Re}(x)$ is continuous. |
The function $f(x, y) = xy$ is continuous on $\mathbb{R} \times \mathbb{R}$. |
If $f$ and $g$ are continuous functions, then so is $f \cdot g$. |
If $f_n$ and $g_n$ are sequences of real numbers such that $f_n \to 0$ and $g_n$ is bounded, then $f_n g_n \to 0$. |
If $f$ is a bounded linear functional on a normed vector space $X$, then $f(x) \to 0$ as $x \to 0$. |
If $f$ is a bounded bilinear function, then $f(x, y) \to 0$ as $y \to 0$. |
If $f$ is a continuous function, then so is the function $x \mapsto c f(x)$, where $c$ is a constant. |
If $f$ is a continuous function, then the function $x \mapsto f(x)c$ is continuous. |
If $f$ is a continuous function on a set $S$, then the function $x \mapsto c f(x)$ is also continuous on $S$. |
If $f$ is continuous on $S$, then the function $x \mapsto f(x)c$ is continuous on $S$. |
The multiplication by a constant function is continuous. |
If $f$ tends to $0$ in $F$, then $f/c$ tends to $0$ in $F$. |
If $f$ tends to $a$, then $f^n$ tends to $a^n$. |
If $f$ tends to $0$ in the filter $F$, then $f^n$ tends to $0$ in the filter $F$, for any positive integer $n$. |
If $f$ is a continuous function, then $x \mapsto f(x)^n$ is continuous. |
If $f$ is a continuous function from $S$ to a real normed algebra, then the function $x \mapsto f(x)^n$ is continuous. |
If $f_i$ converges to $L_i$ for each $i \in S$, then the product $\prod_{i \in S} f_i$ converges to $\prod_{i \in S} L_i$. |
If $f_i$ is continuous for each $i \in S$, then the product $\prod_{i \in S} f_i$ is continuous. |
If $f_i$ is continuous for each $i \in S$, then the product $\prod_{i \in S} f_i$ is continuous. |
The sequence of real numbers $f_n$ converges to $c$ if and only if the sequence of complex numbers $f_n$ converges to $c$. |
The limit of a function $f$ plus a constant $c$ is the constant $c$ plus the limit of $f$. |
If $f$ and $g$ tend to $a$ and $b$, respectively, then $f^g$ tends to $a^b$. |
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