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 $2\Vert f{\Vert }_{\infty }\delta$. 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.