Extreme value theorem

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In calculus, the extreme value theorem states that if a real-valued function ${\displaystyle f}$ is continuous on the closed and bounded interval ${\displaystyle [a,b]}$, then ${\displaystyle f}$ must attain a maximum and a minimum, each at least once. That is, there exist numbers ${\displaystyle c}$ and ${\displaystyle d}$ in ${\displaystyle [a,b]}$ such that: $${\displaystyle f(c)\leq f(x)\leq f(d)\quad \forall x\in [a,b].}$$

The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function ${\displaystyle f}$ on the closed interval ${\displaystyle [a,b]}$ is bounded on that interval; that is, there exist real numbers ${\displaystyle m}$ and ${\displaystyle M}$ such that: $${\displaystyle m\leq f(x)\leq M\quad \forall x\in [a,b].}$$

This does not say that ${\displaystyle M}$ and ${\displaystyle m}$ are necessarily the maximum and minimum values of ${\displaystyle f}$ on the interval ${\displaystyle [a,b],}$ 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.