Check the existence of second order partial derivatives at $(0,0)$ of $$ f(x,y) = \left\{ \begin{array}{rl} y^3 \cos(\frac{1}{x^2 + y^2}), & \text{if } (x,y) \neq 0 \\ 0, & \text{if } (x,y) = 0. \end{array}\right. $$
$f$ is continuous at $(0,0)$, but it does not imply existence of even the first order partial derivative.
How should I approach this problem? By definition?
$$ f_{xy}(0,0) = \lim \limits_{y \to 0} \frac{f_x(0,y)-f_x(0,0)}{y} = \lim \limits_{y \to 0} \frac{0}{y} = 0, $$
because: $f_x(0,0)=0$
$f_x(x,y) = \frac{2xy^3 \sin(\frac{1}{x^2 + y^2})}{(x^2 + y^2)^2}$
$f_x(0,y) = 0$.
Is this correct?