Problem $3\mathrm{.}1(11\mathrm{.}1$: Partial Derivatives$).$ Define the function $f$:$R^{2}Rby$

$f(x,y)=\{\begin{array}{ccc}{x}^{2}sin(\frac{1}{x})+y^{2}cos(\frac{1}{y})& ifx0\text{a}\text{n}\text{d}& y0\\ x^{2}sin(\frac{1}{x})& ifx0,& y=0\\ y^{2}cos(\frac{1}{y})& ifx=0,& y0\\ 0& ifx=0,& y=0\end{array}$

Compute the partial derivatives $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}at(0,0),if$ they exist. Prove that one partial

derivative exists and $is$ continuous $at(0,0),$ while the other exists but $is$ not continuous

$at(0,0).$