Is diffirentiability in finite dimensional space is equivalent to the existence of partial derivatives Let $f$ be a continuous mapping, defined on $\mathbb{R}^2$. I want to prove that $f$ is differentiable at a point $(a,b)\in \mathbb{R}^2$ : Is it sufficient to prove that $f$ has first order derivative at $a$ and $b$ i.e $f_x(a,b)$ and $f_y(a,b)$ exists (since we have a finite dimensional space)?
Is there a reference book I could look at to find such result ?
Define $f(x) = \min(x_1,x_2)$ for $x\ge 0$ and zero otherwise.
$f$ is not differentiable at zero but the partials exist at $x=0$.
$f$ might even fail to be continuous at $(a,b)=(0,0)$ :
$$
f(x,y)=\left\{
\matrix{
0, &amp; \text{ if } x=0 \text{ or } y=0,\hfill\cr
1, &amp; \text{ if } x\neq0 \text{ and } y\neq0.
}
\right.
$$