Hello there, I have this question that comes with a hint; however the hint doesnt seem to make any sense to me whatsoever so I would appreciate some insight into this.

Let f

be a function. We define

0 as

to mean that

such that

Similarly define

as

to mean that

such that

Now suppose that

is continuous and has

as

and

as

. Show that f is bounded and attains its maximal value on

, i.e. there is some

with

.

[HINT: If f(x) = 0 for all x, there is nothing to prove (why not?

[well because f(x)= a constant so it is bounded by that and that is its maximal value for all x] so we can assume that there is some with f(x) > 0. Then take and find such that and for or . Now use the extreme value theorem ] Solution: or as much as I can do...
Now

such that

.

Let

. Choose

such that

.

Choose

such that

.

But

which is a contradiction.

And

which is a contradiction.

Hence

.

But this doesn't seem to show me anything?? Any help with this would be deeply appreciated.