Suppose $ F: \mathbb{R} \to \mathbb{R}$ is a montonic increasing right-sided continuous function. Denote the Borel-Sigma-Algebra as $B$. And define $\mu: B \to [0, \infty]$ as $\mu((a,b)) := F(b) - F(a) $ for all $a,b \in \mathbb{R}$ and $a<b$. I want to show that, if F is on every finite interval piecewise continuously differentiable, and $x_k \in \mathbb{R}$ and let $\delta_{x_k} $ denote the Dirac-Measure. Then we can write: $ \mu(A)= \int_A f d\lambda + \sum_{k=1} ^\infty g_k\delta_{x_k}(A)$ for all $A \in B$, such that $f: \mathbb{R} \to [0, \infty)$ is measurable and $g_k \in [0,\infty)$.
I am not sure what could be meant with piecewise continuously differentiable on every finite interval. I am also struggling to prove it for any Borel set, since the measure $\mu$ is only defined on intervals. I am also wondering if the condition of piecewise continuously differentiable on every finite interval is really necessary. If it is necessary, I would be very glad if someone could show me a counterexample.