Interactive Real Analysis
- part of Examples 6.4.7(a):
Find a continuous function on a bounded interval that is unbounded.
How about a continuous function on a bounded interval that does
not have an absolute maximum but is bounded ?
Back
If the function was defined on a bounded and closed set (i.e.
a compact set), it would have to have an absolute maximum and
minimum, and would therefore be bounded. To construct our counterexample,
we again have to define a function on an open, bounded interval.
The rest is left as an exercise.
BackWhile at first glance confusing, all we have do remember is that if the domain of the function was a compact (i.e. closed and bounded) interval, the function would have to have an absolute maximum. Hence, to construct a counterexample, we need to define a continuous function on an open, bounded interval. The rest is easy, and left as an exercise.
These examples show that the closedness condition in the Max/Min theorem for continuous functions is essential.