$(12$ points$)$ Consider the function

$f(x,y)=\{\begin{array}{ccc}\frac{x^2{y}^2}{x^4+y^4}& \text{f}\text{o}\text{r}(x,y)(0,& 0)\\ k& \text{f}\text{o}\text{r}(x,y)=(0,& 0)\end{array}$

for a constant $k.$

$(a)$ For which $k,if$ any, $isf(x,y)$ continuous $at$ the origin? Justify.

$(b)Ifk=0,$ calculate the gradient $at$ the origin, gradf$(0,0),ifit$ exists. $If$ not, justify.

$(c)$ Let $u()be$ the unit vector pointing $in$ the direction $in$ polar coordinates. Using the

definition $of$ the directional derivative, given $an$ angle $,$ for what value$(s)ofk$ would

$D_{u()}f(0,0)$ exist? What $is$ the value $of{D}_{u()}f(0,0)$ for such $k.$