real analysis - Example for a continuous function that has directional derivative at every point but not differentiable at the origin

Can we find a function $f:\mathbb R^n\to\mathbb R$ that such that $f$ is continuous and $\partial_v f(p)$ exists for all $p\in\mathbb R^n$ and $v\in\mathbb R^n$. But $f$ is not differentiable at $0$?

Is such function $f$ exists?

Here give a example that has directional derivative everywhere, but it's not continuous at the origin.

Answer

Consider the polar coordinates $\ (r\,\ \Phi)\ $ in $\ \mathbb R^2.\ $ Function $ f:\mathbb R^2\rightarrow\mathbb R,\ $ which in polar coordinates is given by:

$$ f(r\,\ \Phi)\,\ :=\,\ r\cdot\sin(3\cdot\Phi) $$

is continuous everywhere, is infinitely differentiable outside the origin $\ O,\ $ and $\ f\ $ has all directional derivative at $\ O,\ $ but $\ f\ $ is not differentiable at $\ O.$