= Lipschitz composition preserves Riemann integrability
{c}
If $f$ is <Riemann integrable> and $g$ is Lipschitz on an interval containing the range of $f$, then $g\circ f$ is <Riemann integrable>. With Lipschitz constant $K$, the oscillation on each partition cell satisfies $\operatorname{osc}(g\circ f)\leq K\operatorname{osc}(f)$, and hence the gap between <Darboux sums> is multiplied by at most $K$. A continuously differentiable $g$ is Lipschitz on each closed bounded interval by the <mean value theorem> and the <extreme value theorem> applied to $g'$.
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