Exercise 6, § 1

Let $(r_n)$ be the sequence formed by the rational numbers in $[0, 1]$, arranged in a certain order. Show that the function $f(x) = \sum_{n=0}^{\infty} 2^{-n}(x - r_n)^{1/3}$ is continuous and differentiable at every point of $\mathbf{R}$, and has an infinite derivative at every point $r_n$. (To see that $f$ is differentiable at a point $x$ distinct from the $r_n$, distinguish two cases, according to whether the series with general term $2^{-n}(x - r_n)^{-2/3}$ has sum $+\infty$ or converges; in the second case, note for all $x \neq 0$ and all $y \neq x$, one has