Is it possible to say that the directional derivatives of a function f at a exists but f is not differentiable at a? If so, why? I cannot get the intuition about it. Could someone please elaborate on this point a little bit?  I am self studying mostly, so I need to discuss these trivial matters with someone :)  Thanks in advance!
Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto   \begin{cases}         0 &amp; \text{for } (x,y)=(0,0) \\    \frac{x^3}{x^2+y^2}       &amp; \text{for } (x,y) \neq (0,0)   \end{cases} $$  Show that $f$ not differentiable at $(0,0)$ but all directional derivatives exist.  I don't know how to tackle this problem. Can someone give some hints or solve the problem? Thanks :)
Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto   \begin{cases}         0 &amp; \text{for } (x,y)=(0,0) \\    \frac{x^3}{x^2+y^2}       &amp; \text{for } (x,y) \neq (0,0)   \end{cases} $$  Show that $f$ not differentiable at $(0,0)$ but all directional derivatives exist.  I don't know how to tackle this problem. Can someone give some hints or solve the problem? Thanks :)