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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