I have always found interesting, as a student as well as teacher, the "proof" that every derivative is continuous:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Fix any $x_0 \in \mathbb{R}$ and $h > 0$, by the mean value theorem we find $\xi \in (x_0,x_0+h)$ such that:
$$ f'(\xi) = \frac{f(x_0+h) - f(x_0)}{h} \implies \lim_{h \to 0} f'(\xi) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h} =f'(x_0),$$
where in the last equality we used that $f$ is differentiable. The conclusion follows since $h \to 0$ entails $\xi \to x_0$.