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?
Can the following theorems be used to prove that every continuous function on a closed interval $[a,b]$ is Riemann integrable?
If so, how?
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.
The standard proof uses the following theorem:
If $f\colon[a,b]\to\mathbb{R}$ is continuous, then it is uniformly continuous.