How to prove from first principles that a function is Riemann integrable

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How to prove from first principles that a function is Riemann integrable

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Quick description

This article is about how to prove that a function is Riemann integrable if for some reason you need to do so directly from the definition.

Prerequisites

Basic undergraduate real analysis.

Example 1

If $f$ and $g$ are both Riemann integrable functions defined on $[a,b]$ , then so is $f+g$ , and $\int_a^b(f(x)+g(x))dx=\int_a^bf(x)dx+\int_a^bg(x)dx$ .

Example 2

Every monotone function defined on the closed interval $[a,b]$ is Riemann integrable.

Example 3

Every continuous function on the closed interval $[a,b]$ is Riemann integrable.

General discussion

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