= Bounded functions continuous away from one endpoint are Riemann integrable
Let $f$ be bounded on $[a,b]$ and continuous on $(a,b]$. For any positive error tolerance, isolate $[a,a+\delta]$. Its contribution to the difference of <upper Darboux sums> and <lower Darboux sums> is at most $2\|f\|_\infty\delta$. On $[a+\delta,b]$, <uniform continuity> supplies a <partition of an interval> with arbitrarily small remaining oscillation sum. The <Riemann integrability criterion> then proves integrability, even when the values oscillate without a limit at the endpoint.
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