Example of non-differentiable continuous function with all partial derivatives well defined

Question

Give an example of a function $f : \mathbb{R}^3 \to \mathbb{R}$ such that the partial derivatives exist at $(0,0,0)$, and $f$ is continuous at $(0,0,0)$, but it is not differentiable at $(0,0,0)$. Any hint?

Answer 1

Consider $f\colon \mathbb R^3\to \mathbb R, (x,y,z)\mapsto \sqrt{|xy|}$.

All the partials at the origin are null.

The limit $\lim \limits_{(x,y,z)\to (0,0,0)}\left(\dfrac{f(x,y,z)-f(0,0,0)-\langle (0,0,0), (x,y,z)\rangle}{\sqrt{x^2+y^2+z^2}}\right)$ isn't $0$ (take the sublimits along the path $t\mapsto (t,t,0)$).