Let $$f(x,y)= \begin{cases} \sin\left(\frac{y^2}{x}\right)\sqrt{x^2+y^2}, & x\neq0\\ 0,&x=0 \end{cases}$$
Then, is $f$ differentiable at the origin? I think no, but I also find that the directional derivatives exists at all points and is equal to 0, which is a linear function. Any hints. Thanks beforehand.
You're correct that all directional derivatives vanish at the origin. (But there are discontinuous functions that have that property!) Here's a hint to answer your question: Is $$\lim_{(x,y)\to (0,0)} \sin\big(\tfrac{y^2}x\big) = 0?$$