Let $f:X \to Y$ be a map between Hilbert spaces. It has a directional derivative $f'$:
$$f(x+th)=f(x) + tf'(x)(h) + r(t)$$ where $f'$ is not necessarily linear, and $\frac{r(t)}{t} \to 0$ as $t \to 0$ (the remainder term).
So $$\lim_{t \to 0} \frac{f(x+th)-f(x) - tf'(x)(h) }{t} = 0$$ holds, but if it holds uniformly in $x \in S$, $S$ is some subset of $X$, then it is said to be uniformly directionally differentiable.
I want to know: if $f$ is directionally differentiable, are there any conditions on $f$ or $f'$ that guarantee uniform directionally differentiable? Assume $f$ is Hadamard differentiable if necessary.
In the case of real-valued functions, I recall something like continuity of $f'$ is enough, but here, I don't know.