Both the absolute and local (or relative) extrema have important theorems associated with them.

Extreme Value Theorem

The extreme value theorem states the following: if f is a continuous function on the closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on [a, b].

For example, it can be seen in the three continuous functions below that f attains both an absolute max and an absolute min on [a, b]:

Figure %: Demonstrating the extreme value theorem on continuous functions

Upon reflection, this theorem should seem intuitively obvious, but it is actually very difficult to prove, so the proof will be omitted here.

Note that the extreme value theorem only applies to continuous functions on a closed interval. If, for example, we had a continuous function on an open interval, the EVT would not apply. Consider the example of the function f (x) = x on the open interval (0, 1):

Figure %: The EVT does not apply to function defined on an open interval.

Note that f (x) does not attain a minimum value on this open interval, since as x approaches 0, f (x) gets smaller and smaller, but never actually reaches 0. Similarly, there is no absolute max, because as x approaches 1, f (x) gets closer and closer to 1, but never actually reaches it.