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.

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Continuity

Real Analysis

Measure Theory

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