Ngô Quốc Anh

April 5, 2015

The set of continuous points of Riemann integrable functions is dense

Filed under: Uncategorized — Ngô Quốc Anh @ 15:03

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

Lemma: Assume that P=\{t_0=a,...,t_n=b\} is a partition of [a,b] such that

\displaystyle U(f,P)-L(f,P)<\frac{b-a}m

for some m; then there exists some index i such that M_i-m_i < \frac 1m where M_i and m_i are the supremum and infimum of f over the subinterval [t_{i-1},t_i].

We now prove this result.

Proof of Lemma: By contradiction, we would have M_i-m_i \geqslant \frac 1m for all i; hence

\displaystyle \frac{b-a}m = \sum_{i} \frac{t_i-t_{i-1}}{m}\leqslant \sum_{i} \big(M_i-m_i\big)(t_i-t_{i-1})=U(f,P)-L(f,P),

which gives us a contradiction.

We now state our main result:

Theorem. Let f be Riemann integrable over [a,b]. Define

\displaystyle \Gamma = \{ x\in [a,b] : f \text{ is continuous at } x\}

Then \Gamma is dense in [a,b].



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