Continuous functions are Riemann integrable

Continuous functions are Riemann integrable: If $f:[a,b]\to\mathbb{R}$ is continuous , then $f$ is Riemann integrable on $[a,b]$.

This theorem guarantees that the Riemann integral covers all standard functions from calculus and is a basic entry point for more refined integrability criteria.

Proof sketch: A continuous function on the compact interval $[a,b]$ is uniformly continuous . Given $\varepsilon>0$, choose $\delta>0$ so that $|x-y|<\delta$ implies $|f(x)-f(y)|<\varepsilon/(b-a)$. Take a partition $P$ with mesh $<\delta$. Then on each subinterval, the oscillation of $f$ is $<\varepsilon/(b-a)$, so $U(f,P)-L(f,P)<\varepsilon$, proving integrability.