Uniform convergence to the derivative

Question

Suppose $\phi \in C^{\infty}(\mathbb{R})$. Then is it true that $$ \frac{\phi(x+h)-\phi(x)}{h} \to \phi'(x) \quad (h \to 0) $$ uniformly on $\mathbb{R}$? It seems to me like this is trivially the case by definition of the derivative...

Answer 1

No, it's not true. Let $\phi(x) = e^x$, then

$$\frac{\phi(x+h)-\phi(x)}{h} -e^x = e^x\left(\frac{e^h-1}{h}-1\right) \approx e^x\frac{h}{2},$$

and for any $h\neq 0$, that difference becomes arbitrarily large for $x\to +\infty$.

The convergence is uniform on every compact subset of $\mathbb{R}$, however.