Directional derivatives do not imply differentiability
= Directional derivatives do not imply differentiability
Define $f(x,y)=x^3/(x^2+y^2)$ away from the origin and zero at the origin. Every <directional derivative> exists. Along a unit vector $n$, the derivative at the origin is $n_1^3$, which is not linear in $n$. Thus these directional derivatives cannot be the restrictions of one <Fréchet derivative>, and the function is not <Fréchet differentiable> at that point.