Continuous differentiability of functions of one real variable

Question

Let $U$ be an open subset of $\mathbb R$ and let $f:U\to\mathbb R$ be a differentiable function. Define the function $F:U\times U\to\mathbb R$ by $$F(x,y)=\begin{cases} \dfrac{f(x)-f(y)}{x-y}&\textrm{ in case }x\ne y\\ f'(x)&\textrm{ in case }x=y. \end{cases}$$ Then I notice that $f'$ is continuous if and only if $F$ is continuous. So this gives a criterion for the continuous differentiability of functions of one real variable. Is there a related criterion for functions of several variables?