Continuity on a set - Imported Knowlpedia Content

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 $f$ is continuous at every point $a\in A$ (equivalently, the restriction $f|_A:A\to Y$ is continuous).

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 $x_n\in A$ 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 $f^{-1}(V)$ is open in $A$ (i.e., $f^{-1}(V)=A\cap U$ for some open $U\subseteq X$ ).

Examples

Examples:

Any polynomial $p:\mathbb{R}\to\mathbb{R}$ is continuous on every $A\subseteq\mathbb{R}$ .
$f(x)=1/x$ is continuous on $(0,\infty)$ but not continuous on a set containing $0$ .

Remarks

Connection: if $f$ is differentiable (see derivative), then $f$ is continuous on its domain.