định lý giá trị trung bình cho tích phânmean value theorem for integrals

The mean value theorem for integrals guarantees that a function continuous on a closed interval equals its average value at least once on that interval. Specifically, if $f$ is continuous on $[a,b]$, then there is a number $c\in[a,b]$ such that $\int_a^b f(x)\,dx=f(c)(b-a)$, that is, $f(c)=\frac{1}{b-a}\int_a^b f(x)\,dx$.

For a positive function, the area under the graph equals the area of a rectangle with base $b-a$ and height $f(c)$. Continuity is required: a function with a jump discontinuity may never equal its average value. For example, let $f(x)=0$ for $x<1$ and $f(x)=1$ for $x\ge1$ on $[0,2]$. Its average value is $0.5$, but $f$ never equals $0.5$. The theorem only guarantees that $c$ exists; it does not say $c$ is unique.