Mean value theorem for integrals

Mean value theorem for integrals: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be continuous. Then there exists $c\in[a,b]$ such that

Equivalently, the average value $\frac{1}{b-a}\int_a^b f(x)\,dx$ is achieved by $f$ at some point; this relies on the intermediate value theorem together with existence of the Riemann integral for continuous functions.