Differentiability of $f$ in $\mathbb{R^m}$

Question

Is true that if $A\subset \mathbb{R^m},$ $f:A\longrightarrow \mathbb{R}$ and $D_jf$ (partial derivatives) exist in and are bounded in a neighborhood of $a\in A$ then $f$ is differentiable?

I think this is true but I don't see how to use the fact that there exist $M>0$ such that $|D_jf|<M$ for each j. Also, I know if I can show that $D_j f$ is continuous at $a,$ the result follows from that.

Thanks for any hint!

Answer 1

This counter example disproves the statement. (Note: Total derivative exist is equivalent as all directional derivatives exist.)