Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 18 / a / i / Solution

Partition the finite interval as $a=x_0<x_1<\cdots <x_N=b$, with $\Delta x_i=x_{i+1}-x_i$. For a Riemann integrable function, the left-endpoint Riemann sums satisfyThe maximum subinterval width, rather than merely the number of subintervals, must tend to zero. Continuity of $f$ on the closed interval is a sufficient integrability hypothesis. More generally, any chosen sample point in each subinterval gives the same definite integral under these assumptions.