Continuity in a point in a metric space

Question

Let's say $f:A \to B$ is a function between metric spaces, and that $f$ is continuous at $a \in A$.

Does it imply that there exists a ball $B(a,r)$ such that $f$ is continuous for every point of $B(a,r)$ ?

I can't think of a counterexample. Thanks!

Answer 1

Isn't $f: \Bbb R \rightarrow \Bbb R \text{ defined by }f(x)=x^2 \text{ for } x \text{ rational}, 0 \text{ for } x \text{ irrational}$, a counterexample?