Erdős Problem #50 — Differentiability of the distribution function of $\varphi(n)/n$

A more serious problem states as follows: Schoenberg proved about 70 years ago that $\varphi(n)$ has a distribution function. In other words the density of the integers for which $\varphi(n)/n < c$ exists for every $c$ ($0 \leq c \leq 1$). I proved that the distribution function is purely singular. Denote the distribution function by $f(x)$. Is it true that for no $x$ can $f(x)$ have a finite positive derivative? (250 dollars for a proof or disproof.)