Differentiability of a multivariable function with directional derivatives existing

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.

• What is the definition of $f$ in points of the form $(0,t)$ ?? – Fred Jan 31 '18 at 7:46
• @Fred edited the post. Please check now – vidyarthi Jan 31 '18 at 7:57

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 $$\lim_{(x,y)\to (0,0)} \sin\big(\tfrac{y^2}x\big) = 0?$$
• I suspected it. If we approach the origin via the parabola $y^2=x$, we obtain that the function is discontinuous. Am I right in that? But, the directional derivative is linear, right? Also, how to best approach such kind of problems? Any standard algorithm? – vidyarthi Jan 31 '18 at 8:05