1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Functions, Mappings and Intervals.

  1. Oct 27, 2007 #1
    1. The problem statement, all variables and given/known data
    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 ?
  2. jcsd
  3. Oct 27, 2007 #2


    User Avatar
    Science Advisor

    [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.
  4. Oct 27, 2007 #3
    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?
  5. Oct 29, 2007 #4
    Anyone :(
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Functions, Mappings and Intervals.
  1. Mapping functions (Replies: 19)

  2. Functions With Intervals (Replies: 11)

  3. Mapping a function (Replies: 18)