Uniform Convergence of Difference Quotients to Partial Derivative

I know that this, is an old question but one can improve the answer to the case mentioned in Evans book. In particular, one can show the uniform convergence of the difference quotients to the derivative for $C_c^1(\mathbb{R}^n)$ functions. The step for the second derivative follows from this case:

The key idea is to use that for $f\in C_c^1(\mathbb{R}^n)$ we find that $\nabla f$ (and also $f$) is uniformly continuous. Hence, we receive with the mean value theoren:

$$| \frac{f(x+he_i)-f(x)}{h} -\partial_i f(x)| = |\partial_i f(y) -\partial_i f(x)| $$ for some $y$ in the line from $x$ to $x+he_i$ and consequently for some $y\in B_h(x)$.

Now let $\varepsilon >0$ be given and $x\in \mathbb{R}^n$ be arbitrary. Then, by the uniform continuity there is a $\delta >0$ (independent of $x$) such that for all $y\in B_\delta(x)$ we find $|\partial_i f(x) -\partial_i f(y)|< \varepsilon$. Choosing $h\leq\delta$ proves the statement.

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.