Any cumulative distribution function on [0,1] is a convex combination of a continuous cdf and a discrete cdf?

Question

Given that any increasing function on [0,1] is the sum of a step function and a continuous function, can we show that any cumulative distribution function on [0,1] is a convex combination of a continuous cdf and a discrete cdf? It would follow that each Borel probability measure on [0,1] is the convex combination of prob measure with continuous density and a discrete probability measure.

Answer 1

One counterexample is the Cantor distribution, a famous distribution that's neither discrete nor continuous nor mixed. In fact, we call it a singular distribution.