Here's a problem my friend proposed.
A function $f(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}$ satisfies the following conditions.
At $(0,0)$, $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous.
There exists an $\epsilon > 0$ such that in a disc $D_\epsilon (0,0)$, $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ exists.
Does there exists an $\epsilon > 0$ such that in a disc $D_\epsilon (0,0)$, $f$ is differentiable?
We tried to use the definition of differentiability, (i. e. using increments) but to no avail. Is there a counterexample, or a valid proof? Thanks in advance.