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