Continuity Must Hold in an Entire Open Set? Claim: If a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ is continuous at $\vec a \in \mathbb{R}^n$, it is continuous in some open ball around $\vec a$.
Is this claim false? In other words, is it possible for a function to be continuous at a single point $\vec a$ only, but not in the points around $\vec a$?   

$$f(x) = \begin{cases} x &\mbox{if } x \in \mathbb Q \\
0 & \mbox{if } x \in \mathbb R \setminus \mathbb Q \end{cases}$$
is continuous at $x = 0$ and discontinuous elsewhere (sequence definition of continuity helps here).
