# Theorems to prove that continuous function on closed interval is Riemann integrable

Can the following theorems be used to prove that every continuous function on a closed interval $[a,b]$ is Riemann integrable?

• Intermediate Value Theorem
• Existence of Extrema
• Rolle's Theorem
• Mean Value theorem

If so, how?

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The Mean Value Theorem (and Rolle, which is a particular case) are about differentiable functions, so they certain have nothing to do with integrability.

Regarding the other two, there exist (obviously not-everywhere-continuous) functions that do not satisfy the intermediate value theorem nor achieve their extrema, yet they are Riemann-integrable. So those two have no bearing on the integrability of continuous functions either.

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The standard proof uses the following theorem:

If $f\colon[a,b]\to\mathbb{R}$ is continuous, then it is uniformly continuous.

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