Extreme value theorem

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 ] . {\displaystyle m\leq f(x)\leq M\quad \forall x\in [a,b].} This does not say that M {\displaystyle M} and m {\displaystyle m} are necessarily the maximum and minimum values of f {\displaystyle f} on the interval [ a , b ] , {\displaystyle [a,b],} which is what the extreme value theorem stipulates must also be the case. The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.