Function non-differentiable with partial derivatives

Question

I have to give an example of function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ non-differentiable in the point $a\in\mathbb{R}^2$ but with both partial derivatives equal to $1$. Any ideas?

Answer 1

Hint: Take $a = (0,0)$ for convenience. Define $f(x,y) = x + y$ on the axes. Then you can define $f$ to be anything you like off the axes, and $\partial f/ \partial x(0,0) = 1 = \partial f/ \partial y(0,0)$ will always hold.