Weak convergence of bounded-variation integrators ($F_j\to F\text{ uniformly},{\sup }_j\mathrm{TV}(F_j)<\infty$)

If continuously differentiable $F_j,F$ converge uniformly on a compact interval and have uniformly bounded total variation of a function, then $\int b{F}_j^{\prime }\to \int b{F}^{\prime }$ 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.