TITLE: Uniform convergence of difference quotient
QUESTION [5 upvotes]: Let $\phi\in C^\infty_c(\mathbb R)$ be a smooth function with compact support.
For $h>0$ define the difference quotient $\phi_h\in C^\infty_c(\mathbb R)$ by $\phi_h(t)=\dfrac{\phi(t+h)-\phi(t)}{h}$.
By definition, for fixed $t\in\mathbb R$, we have $\phi_h(t)\to\phi'(t)$ as $h\to 0$.

Question: Can we conclude, that $\phi_h\to\phi'$ uniformly on $\mathbb R$ (as $h\to 0)$?

Motivation: This is used in a proof of Stone's Theorem on the existence of generators of operator groups I'm trying to understand.

REPLY [3 votes]: More generally, any function with uniformly continuous first derivative satisfies 
$$\|\phi_h-\phi'\|_\infty\leq \omega(|h|),$$
where $\omega$ is any modulus of continuity of $\phi'$ (just recall that $\phi_h(x)$ equals a value of $\phi'$ in a point within a distance $|h|$ from $x$).