Mean value theorem for definite integrals/Riemann/Fact

Mean value theorem for definite integrals

Suppose that ${}[a,b]$ is a compact interval, and let

$f\colon [a,b]\longrightarrow \mathbb {R}$

be a continuous function.

Then there exists some

${}c\in [a,b]$ such that

${}\int _{a}^{b}f(t)\,dt=f(c)(b-a)\,.$

Proof, Write another proof