Functions, Mappings and Intervals.

[qspeechc]

Homework Statement

This is a problem I am just trying to do myself to work out some other problem.

I am trying to prove: f:[a, b] → [a, b]
Given: f is continuous on [a, b], for all x in [a, b] then df/dx < 1 , f(a) ≥ a , f(b) ≤ b.

2. The attempt at a solution

First I proved that f(b) - f(a) ≤ b - a. It is simple, but I can give the proof if you wish.
Now I need to prove that for all x in [a, b] , f(a) ≤ f(x) ≤ f(b), but I have no clue how to do this. I thought of using the Mean Value Theorem somehow, but I don't quite know how. I also thought of showing that f(x) ≤ f(x + dx) for x in [a, b], but I don't know how to do that either. Any help ?

 

[HallsofIvy]

[itex]f(a)\ge a[/itex] and [itex]f(b)\le b[/itex] are easy. You are told that f(a) and f(b) are in [a, b]!

Are you assuming that f is onto [a, b]? Otherwise, neither [itex]f(a)\le f(x)\le f(b)[/itex] not df/dx< 1 is true for all x in [a, b]. Take, for example, a= -1, b= 1, f(x)=x. f(a)= f(b)= 1, but f(x) is less than 1 for all other x. Also df/dx> 1 for x> 1/2.

 

[qspeechc]

[itex]f(a)\ge a[/itex] and [itex]f(b)\le b[/itex] are easy. You are told that f(a) and f(b) are in [a, b]!

I do not need to prove that, that is one of the assumptions.

I want to prove that f maps from the interval [a,b] onto [a,b].

The assumptions are:
1) f is continuous on [a, b]
2) for all x in [a, b] then df/dx < 1 ,
3) f(a) ≥ a , f(b) ≤ b.


What I have shown:
f(b) - f(a) ≤ b - a

Now to prove that f maps from [a,b] to [a,b], then I also need to show that for all x in [a,b] it is true that f(a) ≤ f(x) ≤ f(b), that is, no point in the interval [a,b] can be mapped to a point outside the interval [a,b]. How do I do this?

 

[qspeechc]

Anyone :(