I t that there are functions continuous everywhere but differentiable nowhere.Then let c(t) be one of them. And let μ[t1,t2]=c(t2)-c(t1). So in this case c(t) is not differentiable.
If you take a more advanced course in measure theory you will learn that monotone functions are automatically differentiable almost everywhere with respect to the Lebesgue measure. This of course covers cumulative distributions.