The following is Exercise 7:5.3 from Bruckner's Real Analysis:
Apply Theorem 7.22 to an appropriately chosen function $f$ to prove that there exists an absolutely continuous function $F$ that is nowhere monotonic. That is,for every $c,d \in \mathbb{R}$ such that $a≤c<d≤b$, $F$ is not monotonic on $[c, d]$.
Theorem 7.22 Let $f$ be Lebesgue integrable on $[a, b]$, and let $F(x)= \int_a^x f dλ$ for $x \in [a,b]$. Then $F$ is differentiable at almost every point, and $F'=f$ almost everywhere.
How can I apply Theorem 7.22 to do the exercise? Also there seem to be not an easy function to construct (Section 4). And how part (b) of this question is absolutely continuous and how it is nowhere monotone?
Hint:
Consider a Borel set $E$ such that for any open interval $I\subset \mathbb{R}$, $0<\lambda(E\cap I)<\lambda(I)$. You might've constructed such set $E'$ before for in $[0,1]$ with the desired property for intervals $I\subset[0,1]$. The set $E=\bigcup_{n\in\mathbb{Z}}(E'+n)$ will do.
Notice that $E^c:=\mathbb{R}\setminus E$ has a similar property. Indeed,
$$\lambda(I)=\lambda(I\cap E)+\lambda(I\cap E^c)$$ and since $0<\lambda(E\cap I)<\lambda(I)$, we obtain that $0<\lambda(I\cap E^c)<\lambda(I)$.
Define $F(x)=\int^x_0\Big(\mathbb{1}_E-\mathbb{1}_{E^c}\Big)\,d\lambda$. $F$ is absolutely continuous and $F'=\mathbb{1}_E-\mathbb{1}_{E^c}$ $\lambda$-a.s.. Since $E$ and $E^c$ are dense, no interval of positive length has fixed signed, so $F$ is not monotone in $I$.
If you have not worked on the existence of sets such as $E'$, let me know and I can sketch a construction for you.
Check if $F$ passes muster. Otherwise let me know to either fix or remove this post.