Question

2. Suppose [a, b] is a closed bounded interval. If f : [a, b] → R is a continuous function, then prove f has an absolute minimum on [a, b].

Answer 1

The extreme value theorem states that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a,b] such that:

A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that:

The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum. And these are absoute maximum and minimum because every other value of fuction is <= or >= respectively.