Extreme value theorem

Continuous real function on a closed interval has a maximum and a minimum

In real analysis, the extreme value theorem states that if a real-valued function f {\displaystyle f} is continuous on the closed and bounded interval [ a , b ] {\displaystyle [a,b]} , then f {\displaystyle f} must attain a maximum and a minimum, each at least once. That is, there exist numbers c {\displaystyle c} and d {\displaystyle d} in [ a , b ] {\displaystyle [a,b]} such that: f ( d) ≤ f ( x) ≤ f ( c) ∀ x ∈ [ a , b ] . {\displaystyle f(d)\leq f(x)\leq f(c)\quad \forall x\in [a,b].} The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function f {\displaystyle f} on the closed interval [ a , b ] {\displaystyle [a,b]} is bounded on that interval; that is, there exist real numbers m {\displaystyle m} and M {\displaystyle M} such that: m ≤ f ( x) ≤ M ∀ x ∈ [ a , b ] .

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