Exercise 8, § 2

Consider, on the interval $[0,\ 1]$, a family $(I_{n,p})$ of pairwise disjoint open intervals, defined inductively as follows: the integer $n$ takes all values $\geq 0$; for each value of $n$ the integer $p$ takes the values $1,\ 2,\ \ldots,\ 2^n$; one has $I_{0,1} = ]\frac{1}{3},\ \frac{2}{3}[$; if $J_n$ is the union of the intervals $I_{m,p}$ corresponding to the numbers $m \leq n$, the complement of $J_n$ is the union of $2^{n+1}$ pairwise disjoint closed intervals $K_{n,p}$ ($1 \leq p \leq 2^{n+1}$). If $K_{n,p}$ is an interval $[a,\ b]$ one then takes for $I_{n+1,p}$ the open interval with endpoints $b - \frac{b-a}{3} \left( 1 + \frac{1}{2^n} \right)$ and $b - \frac{b-a}{3.2^n}$. Let $E$ be the perfect set which is the complement with respect to $[0,\ 1]$ of the union of the $I_{n,p}$. Define on $[0,\ 1]$ a continuous real function $f$ which admits a right derivative at every point of $[0,\ 1[$, but fails to have a left derivative at the uncountable subset of $E$ of points distinct from the endpoints of intervals contiguous to $E$ (cf. exerc. 7). (Take $f(x) = 0$ on $E$, define $f$ suitably on each of the intervals $I_{n,p}$ in such a way that for every $x \in E$ there are points $y < x$ not belonging to $E$, arbitrarily close to $x$, and such that $\frac{f(y) - f(x)}{y - x} = -1$.)