Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 12D / a

For a partitionputThe upper and lower Darboux sums areThe upper and lower Darboux integrals areThe bounded function is Riemann integrable when these two numbers agree; their common value is its integral.

Let $f$ be continuous on $[0,1]$. Compactness makes it uniformly continuous. Given $\epsilon >0$, choose $\delta >0$ such thatChoose a partition with mesh smaller than $\delta$. The oscillation $M_j-m_j$ on every subinterval is then at most $\epsilon$, soThe Riemann integrability criterion proves that every continuous function on $[0,1]$ is Riemann integrable.