An absolutely continuous function that isn't one when raised to a fractional power.

Question

Suppose that f is an absolutely continuous function on $[0,1]$. Then for $p > 1$, I can show that $|f|^p$ is also absolutely continuous on $[0,1]$. This follows easily by showing that $|x|^p$ is Lipschitz, and then the result is immediate by composing the two functions.

However, this does not seem to follow for $0 < p < 1$, and so I am wondering if there exists a concrete absolutely continuous function $f$ on $[0,1]$ for which $|f|^p$ is not absolutely continuous. Can anyone supply such an example?

Answer 1

The function $g(x) = x|\sin(1/x)|$, $x \neq 0$ and $f(0) = 0$ is not absolutely continuous but its square $f(x)$ is, hence gives an example of what you are looking for.