Let $\displaystyle n=0,1,2,\ldots$ and $\displaystyle f:\mathbb R^2\to\mathbb R$ defined by

$\displaystyle f(x,y)=\left\{\begin{array}{cl}(x+y)^n\sin\dfrac1{ \sqrt{x^2+y^2}}&(x,y)\ne(0,0).\\[0.5cm]0&(x,y)=(0,0).\end{array}\right.$

How should we choose $\displaystyle n$ such that:

a) $\displaystyle f$ is continuous.

b) $\displaystyle f$ is differentiable.

c) $\displaystyle f$ has continuous partial derivatives on every point on $\displaystyle \mathbb R^2.$

Spoiler: