Uniform Convergence of Difference Quotients to Partial Derivative

Question

I'm currently reading Evans' PDE book. In it he claims that for $f \in C^2_c(\mathbb{R}^n)$ $$\frac{f(x + he_i) - f(x)}{h} \to \frac{\partial}{\partial x_i}f(x)$$ and $$\frac{\frac{\partial}{\partial x_i}f(x + he_j) - \frac{\partial}{\partial x_i}f(x)}{h} \to \frac{\partial^2}{\partial x_jx_i}f(x)$$ uniformly as $h \to 0$.

My question is why must the convergence be uniform?

Thanks in advance.

Answer 1

Using Taylor's formula, you get for the first claim :

$$|\frac{f(x+h e_i)-f(x)}{h} - \frac{\partial f}{\partial x_i}(x)| \leq |h| \sup |D^2f|$$

the supremum being finite and well defined since $f$ is $C^2$ with compact support. That gives uniform convergence.

If you add another order of regularity to your hypothesis, the same argument works for the second part. However, as it is, it doesn't work.