Let $f :\mathbb{R^2} \to \mathbb{R}$ be given. Give sufficient condition for the differentiability of $f$ .

Question

Let $f :\mathbb{R^2} \to \mathbb{R}$ be given. Give sufficient condition for the differentiability of $f$ .

How can this be shown? I found a definition which stated that it would be sufficient to show that if one of the partial derivatives $\partial_kf(x)$ exists at $x_0$ and the other partial derivatives $\partial_jf(x), j=1,2n\dots,n$ exists in the ball $B^n(x,r)$ for some $r>0$ and are continuous at $x_0$, then $f$ would be differentiable at $x_0$. However I think this is a farfetched for this, what alternatives do I have?

Answer 1

There are a lot of sufficient condtions for differentiability. For example, if $f$ is a constant function or, more generally, a polynomial.

On the other hand, the more standard result is that $f$ is differentiable at $p$ if $\frac{\partial f}{\partial x_j}$ exist and are continuous at $p$ for each $1 \leq j \leq n$ (for $f: \mathbb R^n \to \mathbb R$).