States that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a,b] such that:
{\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}{\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}
A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that:
{\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}{\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum.