Let $ f: R^{n} \rightarrow R $ be continuous and let $f$ be partially differentiable for every $x \in R^{n}$ $\verb|\|${$0$}, with $lim_{x\rightarrow0}{\partial_if(x)=0}$ for every $i=1,...,n$. Show that $f$ is differentiable in $0$.
My idea was to show that $\partial_if(x)$ has to be continuous in $0$ due to the limit surrounding the point. And I also think $\partial_if(0)=0$, but I need $\partial_if(x)$ to be continuous as well to conclude differentiation in $0$.
I don't know how to continue from here; is my idea wrong? I could really use some advise!
Hint: Use the MVT to see
$$f(x,y)-f(0,0)= f(x,y)-f(x,0)+f(x,0)-f(0,0)$$ $$ = \frac{\partial f}{\partial y}(x,c)\cdot y + \frac{\partial f}{\partial x}(d,0)\cdot x.$$