Example of a multivariable nondifferentiable function with directional derivatives. Please give an example of a continuous function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ all of whose directional derivatives exist, so for every $\mathbf{v}\in\mathbb{R}^2$
$$df_{\mathbf{x}}(\mathbf{v})=\lim\limits_{t\to 0}\frac{f(\mathbf{x}+t\mathbf{v})-f(\mathbf{x})}{t}$$ exists, however $f$ is not differentiable, say at $(0,0)$.

When $f((r\cos\ t,r\sin\ t)) = r \sin\ (2t)$ where $r>0$ and $0\leq
t<2\pi$ and $f(0,0)=0$, then $$ df\ (\cos\ t,\sin\ t)=\lim_r\ \frac{
f((r\cos\ t,r\sin\ t)) -0 }{r} =\sin\ (2t) $$
Hence all directional derivatives exist. Here assume that $f$ is
differentiable So
$$ df\ -(\cos\ t,\sin\ t) =df\ (\cos\ (\pi +t),\sin\ (\pi+t)) =\sin\
(2t)=df\ (\cos\ t,\sin\ t)$$
which is a contradiction. Hence $f$ is not differentiable.
