= Summable absolute-value cusp series
{title2=$G(x)=\sum_nc_n|x-a_n|$}
Let the locations $a_n$ be distinct and bounded, and let $c_n>0$ with $\sum_nc_n<\infty$. 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, $G'(x)=\sum_nc_n\operatorname{sgn}(x-a_n)$; 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>.
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