Let $\{f_{n}\}$ be a sequence of functions differentiable at $x_0$. Let
$r_{n}(h,x_0) = f_n(x_0+h) - f_n(x_0) - f'_n(x_0)h $
We know that for any fixed $n$, $r_n(h,x_0) = o(|h|)$ as $h \to 0$ from the existence of the derivative at $x_0$.
I'm trying to find a property in the literature that I would call "uniform differentiability at a point", and that I would define as
$sup_n|r_n(h,x_0)| = o(|h|)$
That is, the whole sequence of functions has remainders that converge uniformly to zero at $x_0$.
Has such a property been studied before? If so, what is known about it?
I searched for more information but I can only find the term "uniform differentiability" when the uniformity is with respect to $x$ instead of $n$.
EDIT: Found a paper that describes "uniformly strongly differentiable" sequences of functions. It's analogous to my definition but for strong differentiability.
Uniform equi-countinuity of the sequence $\{f_n'\}$ is a sufficient condition since $$ f_n(x+h)-f_n(x)-hf'_n(x)=h\big(\,f_m'(\xi)-f_n'(x)\big) $$ for some $\xi\in(x,x+h)$.
We say that $\{g_n\}$ is uniformly equicontinuous if for every $\varepsilon>0$, there exists a $\delta$, such that for all $n\in\mathbb N$ $$ |x-y|<\delta\quad\Longrightarrow\quad |g_n(x)-g_n(y)|<\varepsilon. $$