Singular distribution functions have zero derivative almost everywhere ($\mu \perp {\lambda }_1⟹F_{\mu }^{\prime }=0{\lambda }_1\text{-a.e.}$)

For $F_{\mu }(t)=\mu ((-\infty ,t])$, every difference quotient at $x$ is nonnegative and bounded by $4\mu (B(x,2|h|))/{\lambda }_1(B(x,2|h|))$. The spherical derivative of a singular measure vanishes, so the two-sided derivative is zero Lebesgue almost everywhere. A singular distribution function need not be constant: differentiation almost everywhere does not recover increments without absolute continuity of a function of the function.