 Summable absolute-value cusp series

Let the locations $a_n$ be distinct and bounded, and let $c_n>0$ with ${\sum }_n{c}_n<\infty$. The Weierstrass M-test gives locally uniform convergence of $G$, so the uniform limit theorem makes it continuous. The reverse triangle inequality bounds each term's difference quotient by $c_n$. Consequently a uniformly small tail permits passing each one-sided difference-quotient limit through the series. At every point outside the locations, $G^{\prime }(x)={\sum }_n{c}_n\mathrm{sgn}(x-a_n)$; at $a_m$, the right derivative minus the left derivative is $2c_m>0$. Thus the function has precisely the prescribed cusp locations, even when they accumulate at another point where the function remains differentiable.