= Integration error controlled by total variation
{title2=$|\int_a^b(g-h)\,df|\leq\|g-h\|_\infty V_a^b(f)$}
Let $f$ have <bounded variation>, and let $g,h$ be <continuous functions> on $[a,b]$. In any tagged sum for the <Riemann-Stieltjes integral>,
$$
\left|\sum_j(g(\xi_j)-h(\xi_j))(f(t_j)-f(t_{j-1}))\right|
\leq\|g-h\|_\infty\sum_j|f(t_j)-f(t_{j-1})|
\leq\|g-h\|_\infty V_a^b(f).
$$
Taking the <limit> proves the displayed error bound. Thus approximating $g$ uniformly to accuracy $\varepsilon$ controls this integration error by $\varepsilon V_a^b(f)$, even when the integrator is not <differentiable>. For the <piecewise linear function> in <partition example of total variation>, accuracy $0.001$ gives an error at most $0.011$. Total variation supplies exactly the movement budget that an endpoint difference would miss.
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