I know that in functions of more than one variable, the existence of partial derivatives does not guarantee that the function will be continuous. However, can the reverse be stated? i.e. that the continuity of the function implies that the partial derivatives exist?
I have a feeling that the answer is no, just like in functions of one variable where e.g. $\mid x\mid$ is continuous at $x=0$ but does not have a derivative.
Best Answer
Nope. Consider $f\colon \mathbb R^2\rightarrow \mathbb R^2$ where $f(x,y) = |x + y|$ at $(x,y) = (0.0)$. It's not hard to show by a similar argument to the one for continuity of $f(x) = |x|$ doesn't imply differentiability that the partials don't exist.