The set of continuous points of Riemann integrable functions is dense

Something …. really “dense” 🙂

Ngô Quốc Anh

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

Lemma: Assume that $latex P={t_0=a,…,t_n=b}$ is a partition of $latex [a,b]$ such that

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

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

We now prove this result.

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

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

which gives us a contradiction.

We now state our main result:

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

$latex displaystyle Gamma = { xin…

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