Example: Does the function have an absolute maximum and minimum on [-2, 1] ? How about on the interval [0, ) ?

This function does have an absolute maximum and minimum on the interval [-2, 1], as predicted by the Max/Min theorem for continuous functions. The absolute maximum is 1, and the absolute minimum is 1/5.

On the other hand, on the unbounded interval [0, ) the function fails to possess both absolute maximum and minimum. While 1 is still the absolute maximum, there no longer is an absolute minimum.

Thus, the boundedness condition in the Max/Min theorem for continuous functions is essential.