Continuity on a set

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 $x\in A$,

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:

• 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 $0$.

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