Is a smooth (infinitely differentiable) map always continuous?

I am not sure about this. Here is my problem:

I have a smooth map $\displaystyle \phi:W \rightarrow \mathbb{R}^3$, where $\displaystyle W\subseteq\mathbb{R}^3 $is open. For $\displaystyle p\in W$, ive got an open neighbourhood of $\displaystyle \phi(p) \in \mathbb{R}^3$, say $\displaystyle V$.

It is claimed that $\displaystyle \phi^{-1}(V)$ is an open neighbourhood of p. Surely $\displaystyle \phi $ must be continuous for this to be the case? Any help with this would be appreciated!