Mean value theorem for definite integrals/Riemann/Fact/Proof
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Proof
On the compact interval, the function is bounded from above and from below, let and denote the minimum and the maximum of the function. Due to fact they are both obtained. Then in particular for all and so
![{\displaystyle {}x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda0a4b129005203d4c9b056c7aa53a1bbbe2840)
Therefore with some . Due to the Intermediate value theorem there exists a such that .
![{\displaystyle {}d\in [m,M]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97516d73669f6a2b13d597275a309e97d6dadbfb)
![{\displaystyle {}c\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e96620a8d7941d7b882a2148f8703b82848ecad)
