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 $h\to 0$.



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> Question: Can we conclude, that $\phi\_h\to\phi'$ uniformly on $\mathbb R$ (as $h\to 0)$?
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Motivation: This is used in a proof of Stone's Theorem on the existence of generators of operator groups I'm trying to understand.



Uniform convergence of difference quotient
By Taylor's theorem
$$\phi(t+h)=\phi(t)+h\phi'(t)+h^2\phi''(t+u(h,t)h)/2$$
where $0\le u(h,t)\le1$. So
$$\phi\_h(t)=\phi'(t)+h\phi''(t+u(h,t)h)/2.$$
As $\phi''$ is in $C^\infty\_c$ it's pretty clear that $\phi\_h\to\phi'$
uniformly.


