The Exercise
Let $f(x,y)=x$ if $|y|>x^2$
and $f(x,y)=0$ otherwise.
Show that all the directional derivatives of $f$ exist at the origin but there does not exist a linear map $D$ such that $D_vf(0)=D(v)$ for each $v\in R^n$. Show that $lim_{t\to 0}(f(tv)-f(0))/t)$ does not converge to $D_vf(0)$ uniformly over $v\in S^1$.
My Attempt
I have no idea yet about the second and third part of the problem, but I'm working on showing that all the directional derivatives of $f$ exist at the origin.
$D_vf(x)=lim_{t\to 0}(f(x+tv)-f(x))/t$
Take any $v\in R^n$
$D_vf(0,0)=lim_{t\to 0}(f((0,0)+tv)-f(0,0))/t=lim_{t\to 0}f(tv)/t$
How do I know that this limit exists?