Continuous function/Closed bounded interval/Maximum is attained/Fact
Theorem on attained maximum
Let be a closed bounded interval, and let
![{\displaystyle {}[a,b]\subseteq \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/051985a6bcaaf44a6b8aecaac8f90eaa813997e3)
![{\displaystyle f\colon [a,b]\longrightarrow \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bda17a8ca7ea71906b37514dd0342131a206a98)
denote a continuous function.
Then there exists some
such that
![{\displaystyle {}x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda0a4b129005203d4c9b056c7aa53a1bbbe2840)
![{\displaystyle f(x)\geq f(x'){\text{ for all }}x'\in [a,b].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c23570315f34232d102fdd8a619f59cc5980d90e)
This means that the function attains its maximum (and its minimum).