Exercise 3, § 2

a) Let $K$ be the triadic Cantor set (Gen. Top., IV, p. 338), let $I_{n,p}$ be the $2^n$ contiguous intervals of $K$ with length $1/3^{n+1}$ ($1 \leq p \leq 2^n$), and $K_{n,p}$ the $2^{n+1}$ closed intervals of length $1/3^{n+1}$ whose union is the complement of the union of the $I_{m,p}$ for $m \leq n$. Let $\alpha$ be a number such that $1 < \alpha < 3/2$; for each $n$ we denote by $f_n$ the continuous increasing function on $[0, 1]$ which is equal to 0 for $x = 0$, constant on each of the intervals $I_{m,p}$ for $m \leq n$, is affine linear on each of the intervals $K_{n,p}$ ($1 \leq p \leq 2^{n+1}$) and such that $f_d'(x) = \alpha^{n+1}$ on each of the interiors of these last intervals. Show that the series with general term $f_n$ is uniformly convergent on $[0, 1]$, that its sum is a function $f$ which admits a right derivative (finite or not) everywhere in $[0, 1[$, and that one has $f'_r(x) = +\infty$ at every point of $K$ distinct from the left-hand endpoints of the contiguous intervals $I_{n,p}$.

b) Let $g$ be a continuous increasing map of $[0, 1]$ onto itself, constant on each of the intervals $I_{n,p}$ (Gen. Top., IV, p. 403, exerc. 9). If $h = f + g$, show that $h$ admits a right derivative equal to $f_d'(x)$ at every point $x$ of $[0, 1[$.