Given the function $$ g(x,y) = \begin{cases} \frac{xy^3}{2x^4+y^4}, \,\,\, \text{if}\,\, (x,y)\neq(0,0)\,\\ 0, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{else} \end{cases} $$ check whether $g$ is continuous and differentiable at $(0,0)$ or not.
Considering the two restrictions $y = \pm x$, it is seen $g$ is not continuous at $(0,0)$. How about differentiability? Is it related to continuity? The two partial derivatives are both $0$ at $(0,0)$. Still I'm not sure $g$ is differentiable at $0$, since I should consider any directional derivative.
Thanks for your patience and help.
Suppose that $g$ is differentiable at $(0,0)$. Then $g$ is continuous at $(0,0)$.
Conclusion ?