if $f$ Riemann integrable then has a continuity point.

Question

Prove that if $f:[a,b] \to \mathbb R$ be a Riemann integrable function. Using the Nested Cantor Intervals Theorem, prove that there exist $c\in [a,b]$ such that $f$ is continuous in $c$

Answer 1

Hint: $f$ is continuous at $c$ if $$ \lim_{\delta\to 0} \left[ \max_{x \in [c-\delta,c+\delta]} f(x) - \min_{x \in [c-\delta,c+\delta]} f(x) \right] = 0 $$