Can the derivative of an absolutely continuous real function have a simple discontinuity? If $f'$ exists everywhere, then we know that it cannot have any simple discontinuities. But in this case we only know that $f'$ exists a.e. (since $f$ is absolutely continuous).

More specifically, I've also been assuming that $f'$ is increasing (on the set where it is defined). In this case, $f'(x+)$ exists everywhere, and I've been asking if there can be a point $x$ such that $f'(x)$ exists but $f'(x)&lt;f'(x+)$. (We know this cannot happen if $f$ is differentiable on $[x, x+\epsilon)$.)
No, $f'(x)$ can  not have simple discontinuities. 

Theorem

Let $f(x)$ be differentiable on some interval $(a,b)$, and continuous on $[a,b]$. 

If $\lim_{x \to a^+}f'(x)$ exists, then it is equal to $f'_+(a)$ (the right derivative), same for the left derivative. (I will prove this bellow)



Consequence:

Because a simple discontinuity at $c\in(a,b)$ implies existence of sideways limits $\lim_{x \to c^+}f'(x)$ and $\lim_{x \to c^-}f'(x)$, and because $f$ is differentiable at $c \ $ $\left(f'_+(c) = f'_-(c) = f'(c)\right)$, it follows from the theorem above that

$$\lim_{x \to c^+}f'(x) = f'_+(c) = f'(c)  = f'_-(c) = \lim_{x \to c^-}f'(x)$$

And thus $f'$ cannot have a simple discontinuity.



Proof of theorem stated above:
$$
f'_+(a) = \lim_{h \to 0^+}\frac{f(a + h) - f(a)}{h},
$$

Now if you apply the MVT on the interval $[a,a+h]$,

\begin{equation}
\frac{f(a + h) - f(a)}{h} = f'(a +h\theta)
\end{equation}

where $\theta \in (0,1)$

Because $\lim_{x \to a^+}f'(x) = L$ exists,

$$
\lim_{h \to 0^+}f'(a +h\theta) = L = \lim_{h \to 0^+}\frac{f(a + h) - f(a)}{h} = f'_+(a)
$$
I believe that $f(x) = |x|$ is a counterexample. Derivative exists except at $x = 0$, but is discontinuous there.