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Real Analysis: Properties of Continuity

  1. Nov 9, 2008 #1
    1. The problem statement, all variables and given/known data

    Suppose f is continuous on [0,2]and thatn f(0) = f(2). Prove that there exists x,y in [0,2] such that |y-x| = 1 and f(x) = f(y)

    2. Relevant equations

    3. The attempt at a solution

    I got the following 1 line proof.

    Suppose g(x) = f(x + 2) - f(x) on I = [0,2]

    this proofs that |x - y| = 1 for x = 1, y = 2

    and f(x) = f(y)

  2. jcsd
  3. Nov 9, 2008 #2
    As you have defined g, unless you know more about f, then g is only defined at 0 and no other points in [0,2]. I mean g(1)=f(1+2)-f(1)=f(3)-f(1), but what is f(3)?

    And how does what you have written show that f(1)=f(2) certainly there are continuous functions on [0,2] with f(0)=f(2) but that do not satisfy f(1)=f(2).

    What do you know about the function h(x)=f(x+1)-f(x)?
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