Continuity on a subset vs continuity on a subspace.

Question

Suppose $f:X\to Y$ is a function between two metric spaces such that $f:X\to Y$ is continuous on $A$ i.e. at each point of $A$.But is this equivalent to saying $f|_A:A\to Y$ is continuous between the two metric spaces $A$ and $Y$.The forward part is true because if $f$ is continuous on $A$,then $f|_A$ is continuous map between $A$ and $Y$.But Suppose $f|_A$ is continuous map between $A$ and $Y$ metric space.This does not imply the first one.For example $f(x)=0$ if $x\neq 0$ and $f(0)=1$,it is not continuous at $0$ ,so $f:\mathbb R \to \mathbb R$ is not continuous on $\{0\}$,because it is not continuous at every point of $\{0\}$.But $f|_{\{0\}}:\{0\}\to \mathbb R$ is continuous map from $\{0\}$ to $\mathbb R$.Does this example work?

Answer 1

Yes, your example works.

$f: \mathbb{R} \to \mathbb{R}$ is not continuous on $\{0\}$ since

$$\lim_{x \to 0} f(x)=0 \neq 1 = f(0)$$

But the resticted map $f\vert_{\{0\}}$ is continuous, for example because it is constant.

So the two statements are not equivalent.