Mean value theorem for definite integrals/Riemann/Fact/Proof

Proof

On the compact interval, the function $f$ is bounded from above and from below, let $m$ and $M$ denote the minimum and the maximum of the function. Due to fact, they are both obtained. Then, in particular, $m\le f(x)\le M$ for all $x\in [a,b]$, and so

$m(b-a)\le {\int }_a^{b}f(t)dt\le M(b-a).$

Therefore, ${\int }_a^{b}f(t)dt=d(b-a)$ with some $d\in [m,M]$. Due to the Intermediate value theorem, there exists a $c\in [a,b]$ such that $f(c)=d$.

To fact