Note: Here all functions are $\mathbb R \to \mathbb R$. $Id$ denotes the identity function.
Let $g_i$ be a family of functions indexed by some (potentially uncountable) index set $I$. Given a function $f$, we say $g_i$ are uniformly $o(f)$ if for every $e > 0$ there exists some $d > 0$ such that $|g_i(x)| \le |ef(x)|$ for all $i$ in $I$ and $x$ in $B_d (0)$.
Call a function $f$ uniformly differentiable if for every $x$ in $\mathbb R$, $f(x+h) = f(x) + L_x (h) + r_x (h)$ for some linear functions $L_x$ and some uniformly $o(Id)$ functions $r_x$.
When is a differentiable function uniformly differentiable?