# 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

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

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 $[a,b]$.