A similar question was asked before (however, there were a few issues with the definitions and answer given, as I pointed out over there): Can a function be differentiable but not strongly differentiable (Knuth)?
I am looking for an example of a strongly differentiable function that is not continuously differentiable. To clarify, a function $f:(a,b) \to \mathbf{R}$ is said to be strongly differentiable at $x$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that $h \mapsto f(x+h)-f(x)-f'(x) h$ is $\varepsilon$-lipschitzian on $|h| < \delta.$ The same definition applies if $f:\mathrm{A} \to \mathrm{W}$ is a function between the open set $\mathrm{A}$ of a normed space with values in some normed space $\mathrm{W}.$
In the linked post, the family of functions $f_\alpha(x) = x^\alpha \sin x^{-1}$ can be shown to be differentiable for $1 < \alpha \leq 2$ but not strongly differentiable, and they can be shown to be continuously differentiable for $2 < \alpha$ (and therefore, strongly differentiable as well since $\mathscr{C}^1$ implies strong differentiability: Show that a function is strongly differentiable if it is continuously differentiable.). But I suppose the concept of strong differentiability is weaker than differentiability with continuity, but I cannot find nor construct any example.