Differentiability of jump functions

Let
$$j_n(x) =  \left\lbrace \begin{matrix}{} 0 & \text{if } x < x_n, \\ \theta_n & \text{if } x = x_n, \\ 1 & \text{if } x > x_n , \end{matrix} \right.$$ For some $0\leq \theta_n \leq 1$,  then the jump function is defined as
$$J(x) = \sum_{n=1}^\infty \alpha_n j_n(x).$$ with $\sum_{n=1}^\infty \alpha_n < \infty$.
Theorem.  If  $J$ is the jump function, then $J'(x)$ exists and vanishes almost everywhere.  (non-zero in a set of measure zero, $E = \{x : J'(x)\neq 0, x\in \mathcal{B} \}, m(E) = 0$ ).

Typical a probability distribution $F$ is defined as a nondecreasing, right continuous function with $F(-\infty) = 0,\; F(\infty)=1$.