= Weak convergence of bounded-variation integrators
{title2=$F_j\to F\text{ uniformly},\quad\sup_j\operatorname{TV}(F_j)<\infty$}
If continuously differentiable $F_j,F$ converge uniformly on a <compact> interval and have uniformly bounded <total variation of a function>, then $\int bF_j^{\prime}\to\int bF^{\prime}$ for every continuous $b$. For continuously differentiable test functions this is <integration by parts>. Uniform approximation by <polynomials> extends the conclusion to continuous tests using the variation bound.
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