= 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> satisfy
$$
\boxed{\int_a^b f(x)\,dx
=\lim_{\max_i\Delta x_i\to0}\sum_{i=0}^{N-1}f(x_i)\Delta x_i.}
$$
The 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.
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