Continuity on a set - Knowlpedia

Let $f:(X,d_X)\to(Y,d_Y)$ be a function between metric spaces, and let $A\subseteq X$ . We say $f$ is continuous on $A$ if the restriction $f|_A:A\to Y$ , with $A$ carrying the subspace metric, is continuous. Equivalently, $f|_A$ is continuous at every point $a\in A$ .

Spelled out: for every $a\in A$ and every $\varepsilon>0$ , there exists $\delta>0$ such that for all $x\in A$ ,

d_X(x,a)<\delta \quad\Rightarrow\quad d_Y\!\bigl(f(x),f(a)\bigr)<\varepsilon.

Equivalent characterizations

Equivalent viewpoints (metric spaces):

Sequential: if $x_n\in A$ , $a\in A$ , and $x_n\to a$ , then $f(x_n)\to f(a)$ (see limit of a sequence).
Open-set: for every open $V\subseteq Y$ , the preimage $(f|_A)^{-1}(V)=A\cap f^{-1}(V)$ is open in $A$ (i.e., it equals $A\cap U$ for some open $U\subseteq X$ ).

Examples

Any polynomial $p:\mathbb{R}\to\mathbb{R}$ is continuous on every $A\subseteq\mathbb{R}$ .
The function $f:\mathbb R\setminus\{0\}\to\mathbb R$ , $f(x)=1/x$ , is continuous on every subset of its domain.

Remarks

If $f$ is differentiable (see derivative) on an open set, then it is continuous there.