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Mean value theorem for integrals

A continuous function attains its average value somewhere on the interval.

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

∫abf(x) dx=f(c) (b−a).\int_a^b f(x)\,dx = f(c)\,(b-a).
Equivalent characterizations

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