Let $(G,\cdot_G)$ and $(H,\cdot_H)$ be groups with identity elements $1_G$ and $1_H$, respectively. Let $\varphi:G\to H$ be a [group homomorphism](/page/Group%20Homomorphism), meaning that \begin{align*} \varphi(a\cdot_G b)=\varphi(a)\cdot_H\varphi(b) \end{align*} for all $a,b\in G$. Then \begin{align*} \varphi(1_G)&=1_H, \\ \varphi(g^{-1})&=\varphi(g)^{-1} \end{align*} for every $g\in G$, where $g^{-1}$ denotes the inverse of $g$ in $G$ and $\varphi(g)^{-1}$ denotes the inverse of $\varphi(g)$ in $H$.

Let $G$ be a group with centre \begin{align*} Z(G)=\{g\in G: gx=xg \text{ for every } x\in G\}. \end{align*} Let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$, and let $\operatorname{Inn}(G)\le \operatorname{Aut}(G)$ be the subgroup consisting of automorphisms of the form $x\mapsto gxg^{-1}$ for some $g\in G$. Then \begin{align*} G/Z(G)\cong \operatorname{Inn}(G). \end{align*}

Let $G$ be a group with multiplication written multiplicatively. For each $a \in G$, define the conjugation map \begin{align*} \iota_a:G&\to G\\ x&\mapsto axa^{-1}. \end{align*} Then the function \begin{align*} \Phi:G&\to \operatorname{Aut}(G)\\ a&\mapsto \iota_a \end{align*} is a [group homomorphism](/page/Group%20Homomorphism), where $\operatorname{Aut}(G)$ is the automorphism group of $G$ under composition.

Let $(G,\cdot)$ be a group, and let $\operatorname{Aut}(G)$ denote the set of all group automorphisms $\varphi: G \to G$. Then $\operatorname{Aut}(G)$ is a group under function composition.

Let $(X,\tau_X)$ be a topological space, let $Y$ be a set equipped with the indiscrete topology $\tau_Y=\{\varnothing,Y\}$, and let $f:X\to Y$ be a function. Then $f:(X,\tau_X)\to (Y,\tau_Y)$ is continuous.

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, and let $\tau_X$ and $\tau_Y$ be the metric topologies induced by $d_X$ and $d_Y$, respectively. Let $f: X \to Y$ be a map, and let $x_0 \in X$. Then $f$ is continuous at $x_0$ with respect to the topologies $\tau_X$ and $\tau_Y$ if and only if for every $\varepsilon > 0$ there exists $\delta > 0$ such that, for every $x \in X$, \begin{align*} d_X(x,x_0)<\delta \implies d_Y(f(x),f(x_0))<\varepsilon. \end{align*}

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces. Let $A \subset Y$, and equip $A$ with the [subspace topology](/page/Subspace%20Topology) \begin{align*} \tau_A := \{A \cap V : V \in \tau_Y\}. \end{align*} Let $f: X \to A$ be a function, and let $i: A \to Y$ be the inclusion map, defined by $i(a)=a$ for every $a \in A$. Then $f: (X,\tau_X) \to (A,\tau_A)$ is continuous if and only if $i \circ f: (X,\tau_X) \to (Y,\tau_Y)$ is continuous.

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces, and let $f: X \to Y$ be a function. Then $f$ is continuous, meaning that $f^{-1}(V) \in \tau_X$ for every $V \in \tau_Y$, if and only if $f$ is continuous at every point $x_0 \in X$, meaning that for every $V \in \tau_Y$ with $f(x_0) \in V$, there exists $U \in \tau_X$ such that $x_0 \in U$ and $f(U) \subset V$.

Let $(X,\tau_X)$ and $(Y,\tau_Y)$ be topological spaces, and let $\mathcal{B}_Y \subset \tau_Y$ be a basis for the topology $\tau_Y$. For a function $f: X \to Y$, the following are equivalent: 1. $f$ is continuous. 2. For every basis element $B \in \mathcal{B}_Y$, the preimage $f^{-1}(B)$ belongs to $\tau_X$.

Let $(X,\tau_X)$ be a first-countable [topological space](/page/Topological%20Space), let $(Y,\tau_Y)$ be a topological space, and let $f:X\to Y$ be a function. Then $f$ is continuous if and only if, for every point $x\in X$ and every sequence $(x_k)_{k\in\mathbb N}$ in $X$, convergence $x_k\to x$ in $(X,\tau_X)$ implies convergence $f(x_k)\to f(x)$ in $(Y,\tau_Y)$.