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 \(F(-\infty) = 0,\; F(\infty)=1\).