Suppose $\mu$ is a non-atomic measure on the Borel subsets of $[0, 1]$ such that $\mu$ and Lebesgue measure are mutually singular. Show that if $F$ is the cumulative distribution function of $\mu$, defined by $F(x) = \mu([0, x])$ for every $x$ in $[0,1]$, then
$$ \limsup_{h\rightarrow 0^+}\frac{F(x+h)-F(x)}{h}=\infty $$
except possibly on a set of $\mu$-measure $0$.