Let $X \subset \mathbb R^n$, $f:X\to\mathbb R^m$, $x_0\in X$
Assumption: All partial derivatives of f at $x_0$ exist and are continuous
$\Rightarrow$ f is differentiable at $x_0$.
$\Rightarrow D_vf(x_0)=\nabla f(x_0)\cdot v$ (assuming $m=1$ for simplicity)
Which means that all directional derivatives of f at $x_0$ can be expressed as linear combination of the (continuous) partial derivatives of f at $x_0$
Therefore these directional derivatives also have to be continuous. (*)
Is the conclusion correct? (Or why not?) And if yes, is my proof correct? (Or why not?)
(*) I'm implicitly assuming that differentiability at $x_0$ implies differentiability at all points around $x_0$ if they are close enough. Only with this assumption I can conclude that the partial derivatives are defined around $x_0$ and therefore ask if they are continuous around $x_0$ or not.
I hope you can follow my thoughts. Else just ask for clarifications, it's my first question. Thank you for your help :)