If $D_vf(a)=h^2+k^2$,Prove that $f$ is not differentiable on $a$

Question

Assume that for $f:U\subseteq\mathbb R^2\to\mathbb R$ on the point $a\in U$ and in the direction $v=(h,k)$ we have: $D_vf(a)=h^2+k^2$

Prove that $f$ is not differentiable on $a$.

My try:

Assume that $f$ is differentiable on $a$. So, we conclude that $Df(a)(v)=D_vf(a)=h^2+k^2$

But, this doesn't reach a contradiction... That's where i'm stuck...

Answer 1

Hint: prove that, if $f$ is differentiable at $a$, then the map $v\mapsto D_v f(a)$ is linear.