Let

be a function that maps closed and bounded intervals onto closed and bounded intervals, hence if

, then for some real numbers

.

1)Prove that there is an

contained in

such that for all

contained in

2)Prove the range of

has the intermediate value propery.

3)Give an example showing

not needing to be continuous.

My attempts/observations...

1) It would appear that we are looking for an upper bound for

? I'm not reall too sure how to start that one though...

2) Well, possibly we can define

and

and then use that for

,

. Then show that

is between

???

3)I think I can probably take a better guess once I have a better understand of 1) and 2)

Any help would be greatly appreciated.