In this note, we prove that the set of continuous point of Riemann integrable functions on some interval is dense in . Our proof start with the following simple observation.

Lemma: Assume that is a partition of such thatfor some ; then there exists some index such that where and are the supremum and infimum of over the subinterval .

We now prove this result.

Proof of Lemma: By contradiction, we would have for all ; hencewhich gives us a contradiction.

We now state our main result:

Theorem. Let be Riemann integrable over . DefineThen is dense in .

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