I am reading an article and need to prove the following proposition, which is used in the paper:
If a differentiable function $f \colon \mathbf{R} \to \mathbf{R}$ is convex, the derivative $f'$ is monotonically increasing and continuous.
I could prove the monotonicity like this.
It holds from the definition of convexity,
$$
f(rx_1+(1-r)x_3) \le rf(x_1)+(1-r)f(x_3)
$$
for $x_1, x_3 \in \mathbf{R}$ and $r \in (0, 1)$ (and we assume $x_1 < x_3$ here).
Let $x_2 = rx_1+(1-r)x_3$, then
$$
\begin{align}
\frac{f(x_2)-f(x_1)}{x_2-x_1} &\le \frac{(1-r))(-f(x_1)+f(x_3))}{(1-r)(-x_1+x_3)} \\
&=\frac{f(x_3)-f(x_1)}{x_3-x_1}
\end{align}
$$
As $x_2 \to x_1+0$, we obtain
$$
f'(x_1) \le \frac{f(x_3)-f(x_1)}{x_3-x_1}
$$
because the right-hand derivative at $x_1$ is identical to the derivative due to differentiabllity.
We also get
$$
f'(x_3) \ge \frac{f(x_3)-f(x_1)}{x_3-x_1}.
$$
These two inequalities result in $f'(x_1) \le f'(x_3)$, which proves the monotonicity.
However, I cannot prove the continuity of $f'$ by myself. Can anyone prove this?
Correcting my idea above is also appreciated if I have some misunderstanding there.
Best Answer
Continuity follows immediately from the Darboux's Theorem.
It is a straightforward exercise to show that any monotonic function which has the intermediate value property is continuous. Let me know if you need a hint.