Question Number 36176 by prof Abdo imad last updated on 30/May/18
$${let}\:{f}\left({x},{y}\right)\:=\left({x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} \right){sin}\left\{\:\frac{\mathrm{1}}{\:\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }}\right\}\:{if}\left({x},{y}\right)=\left(\mathrm{0},\mathrm{0}\right) \\ $$$${and}\:{f}\left(\mathrm{0},\mathrm{0}\right)=\mathrm{0} \\ $$$${prove}\:{that}\:{f}\:{is}\:{differenciable}\:{at}\:{all}\:{point}\:{of}\:{R}^{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right)\:{prove}\:{that}\:\frac{\partial{f}}{\partial{x}}\:{and}\:\frac{\partial{f}}{\partial{y}}\:{are}\:{not}\:{differdnciable} \\ $$$${at}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$