How to show that the function:
$$f(x,y)= \begin{cases} (x^2+y^2)\sin\frac{1}{(x^2+y^2)^{1/2}}& (x,y)\neq(0,0) \\
0& (x,y)= (0,0)
\end{cases}$$
is differentiable everywhere?
I have been trying to prove it with the definition of differentiability but don't know how cancel out the $x^2$ and $y^2$.
Best Answer
Clearly the partial derivatives exist and are continuous at any point $\;(x,y)\neq(0,0)\;$ and thus the function's differentiable there. At the origin the partial derivatives also exist and equal zero, yet they aren't continuous, so we go by the definition:
$$\frac{f(y,k)-f(0,0)-f'_x(0,0)-f'_y(0,0)}{\sqrt{h^2+k^2}}=\frac{\left(h^2+k^2\right)\sin\frac1{\sqrt{h^2+k^2}}}{\sqrt{h^2+k^2}}=$$
$$=\sqrt{h^2+k^2}\sin\frac1{\sqrt{h^2+k^2}}\xrightarrow[(h,k)\to(0,0)]{}0$$
and thus...