Example of absolutely continuous function $f$ with $\sqrt{f}$ not absolutely continuous
I believe $f(x) = x^2 (\cos \frac1x)^4$ is an example on the interval $(0,1)$. While a proof is certainly needed, the key observation is that sum of the infinitely many local maxima of $f$ converges (indeed $f'$ is uniformly bounded), but the sum of the infinitely many local maxima of $\sqrt f$ does not converge.