# Extreme value theorem

Suppose $f$ is a continuous function on a closed interval $[a,b]$ (note that $f$ may be defined on a bigger domain, but we are interested in the restriction of $f$ to the closed interval and require it to be continuous). Then, $f$ attains its minimum and maximum value on the interval. In other words, the following statements are true:
1. Existence of maximum value: There is a real number $M$ such that $f(x) \le M$ for all $x \in [a,b]$ and there exists $c_1 \in [a,b]$ such that $f(c_1) = M$.
2. Existence of minimum value: There is a real number $m$ such that $f(x) \ge m$ for all $x \in [a,b]$ and there exists $c_2 \in [a,b]$ such that $f(c_2) = m$.