Is there an absolutely continuous function $f\in L^2$ for which $f'\not\in L^2$?

Question

Let $a<b$. We say that $f:[a,b]\to\mathbb C$ is absolutely continuous if there is a $h\in L^1((a,b);\mathbb C)$ with $$f(x)-f(a)=\int_a^xh\;\;\;\text{for all }x\in[a,b].$$ This is equivalent to $f$ being differentiable Lebesgue almost everywhere in $(a,b)$, $$\int_a^b|f'|<\infty$$ and $$f(x)-f(a)=\int_a^xf'\;\;\;\text{for all }x\in[a,b].$$

Is there an example of an absolutely continuous $f\in L^2([a,b];\mathbb C)$ for which $f'\not\in L^2((a,b);\mathbb C)$?

Answer 1

Take $h(t)=\frac 1 {\sqrt t}$ on $[0,1]$ so that $f(x)= 2 \sqrt x$.