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Suppose $f:[a,b] \rightarrow \mathbb{R}$ is a bounded and Riemann integrable function on [c,b] for every c in (a,b). Prove that f is Riemann integrable on all [a,b].

I am trying to use the Darboux criterion to solve this problem. For a given epsilon, I choose a partition of the interval [c,b] and try to relate it to a partition of the interval [a,b]. The problem is that the sup and inf of f may change when we add [a,c] to our partition. Also, by the same token, I cannot use a limit argument to send c to a. Any help would be greatly appreciated.

Tim
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1 Answers1

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A function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for any $\varepsilon >0$ there exists a partition $\mathcal{P}$ of $[a,b]$ such that $$U_{[a,b]}(\mathcal{P},f)-L_{[a,b]}(\mathcal{P},f)<\varepsilon$$ where $U$ and $L$ denote the upper and the lower Riemann sums.

Let $M=\sup |f(x)|.$ For fixed $\varepsilon >0$ let $c$ be chosen so that $c\le a+{\varepsilon\over 4M}.$ By assumptions there is a partition $\mathcal{P}_0$ of $[c,b]$ such that $$U_{[c,b]}(\mathcal{P}_0,f)-L_{[c,b]}(\mathcal{P}_0,f)<{\varepsilon\over 2}$$ Then for $\mathcal{P}=\{a\}\cup \mathcal{P}_0$ we have $$U_{[a,b]}(\mathcal{P},f)-L_{[a,b]}(\mathcal{P},f)\\ \le {\varepsilon\over 2}+U_{[c,b]}(\mathcal{P}_0,f)-L_{[c,b]}(\mathcal{P}_0,f)<\varepsilon $$

Ryszard Szwarc
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  • Technically that's Darboux integrable. It turns out to be equivalent to Riemann integrable, but I personally prefer Darboux as it is often easier to argue about. – Arthur Jun 20 '23 at 06:16