Constructing partition to show that lower sum and upper sum differ by less than $epsilon$
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Let $f$ be a continuous, increasing function on $[a,b]$ . I know that because $f$ is continuous, I can use epsilon-delta to prove that the lower sum $L(f,P)$ and upper sum $U(f,P)$ both converge given a partition of $[a,b]$ : I can divide it equally into $n$ segments, such that the function differs by at most $frac{epsilon}{b-a}$ where the specific length of each section is determined by $delta$ from the epsilon-delta definition. (I think this explanation is correct; if not, please let me know). However, my main question is does this result hold for a non-continuous strictly increasing function on $[a,b]$ ? If so, how would we prove that? Additionally, is there an explicit way to construct a partition (based on $epsilon, b, a$ ) such that it is always the case for any strictly increasing
