jdinatale said:
I've prove everything except for the fact that E is bounded below. It would appear that you would need to know something about the functions taking a minimum value as well to show this using my method, so perhaps there is another way of thinking about things to show a lower bound?
When a continuous function has a maximum value, it means that it reaches the highest possible output within its specified domain. In other words, there is no other point in the function that produces a higher output.
A compact set is a set that is closed and bounded. This means that every sequence of points within the set has a limit point that is also within the set. In simpler terms, a compact set is a set that is finite in size and contains all of its boundary points.
If a continuous function is bounded, it means that its values do not exceed a certain limit. This is important because it ensures that the function does not "blow up" and produce infinitely large values. In order for a continuous function to achieve a maximum, it must be bounded.
No, a continuous function cannot achieve a maximum on an unbounded set. This is because an unbounded set does not have a finite upper limit, which means that the function can continue to increase without ever reaching a maximum value.
Showing that a continuous function achieves a maximum implies compact is important because it helps to prove the existence of a maximum value for the function. This is useful in many mathematical and scientific applications, as it allows us to determine the highest possible output of a function within a given domain.