= Riemann integrability is closed under addition and maximum
{c}
For a common interval partition, the <oscillation of a function on an interval> satisfies $\operatorname{osc}_I(f+g)\le\operatorname{osc}_I f+\operatorname{osc}_I g$. Refining two good <Darboux sums> therefore proves integrability of the sum. <Lipschitz composition preserves Riemann integrability>, so absolute values preserve it too. The identity $\max(f,g)=(f+g+|f-g|)/2$ then proves the assertion for the maximum.
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