# Math Help - Smooth = continuous?

1. ## Smooth = continuous?

Is a smooth (infinitely differentiable) map always continuous?

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

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

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

2. In order to be once differentiable, a function must be continuous.

3. Thanks!