Quadratic damping restores differentiability at the origin ($|f(x)|\le C|x{|}^2⟹Df(0)=0$)

A function with $f(0)=0$ and the displayed bound has zero Fréchet derivative at zero, since $|f(x)|/|x|\to 0$. In particular, a bounded oscillatory factor multiplied by a quadratic expression is differentiable at the origin even if the oscillations have no limit themselves. For $f(x,y)=(x^2-y^2)\sin [1/(x^2-y^2)]$, extended by zero on the two diagonal lines, this damping works at the origin. At a nonzero point of either line, the quadratic factor instead vanishes to first order along a transverse direction; the difference quotient oscillates without a limit. Thus the differentiability set is the complement of those two lines together with the origin.