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Intermediate Value Theorem question

  1. Jul 5, 2017 #1
    Consider a continuous function [itex]f[/itex] in [itex][a,b][/itex] and [itex]f(a) < f(b)[/itex]. Suppose that [itex]\forall s \neq t[/itex] in [itex][a,b][/itex], [itex]f(s) \neq f(t)[/itex]. Proof that [itex]f[/itex] is strictly increasing function in [itex][a,b][/itex].

    2. Relevant equations

    I.V.T: If [itex]f[/itex] is continuous in [itex][a,b][/itex] and [itex]\gamma[/itex] is a real in [itex][f(a),f(b)][/itex], then there'll be at least one [itex]c[/itex] in [itex][a,b][/itex] such that [itex]f(c) = \gamma[/itex].

    3. The attempt at a solution

    This exercise is very strange to me. Besides I can apply the I.V.T to show that for any sub interval in [a,b] there will be an intermediate value in [itex]f(a), f(b)[/itex], I can easily draw and counter example of what it pretends:
    https://www.dropbox.com/s/dtj28xo4ilaai4z/pf.eps?dl=0

    I am missing something important?

    Thanks in advance!
     
  2. jcsd
  3. Jul 5, 2017 #2

    Ray Vickson

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    Your example violates the hypotheses of the claim: your function ##f(x)## has ##f(s) = f(t)## for several pairs ##(s,t)## with ##s \neq t##.
     
  4. Jul 5, 2017 #3

    BvU

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    Picture doesn't make it. Blank screen. Can you copy/paste it in the post ?
     
  5. Jul 5, 2017 #4
    upload_2017-7-5_11-24-35.png
     
  6. Jul 5, 2017 #5
    I did the first one in libreoffice, so I made the mistake pointed by Ray.
     
  7. Jul 5, 2017 #6

    Mark44

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    Your drawing violates the assumption that ##f(s) \neq f(t)##
    Minor point. The verb is "to prove". The noun is "proof".
     
  8. Jul 5, 2017 #7
    Thank you! Now I got the point! Sorry for my poor English!! Thank you very much. Now it's clear for me!
     
  9. Jul 5, 2017 #8

    Mark44

    Staff: Mentor

    No need for an apology. Lots of native speakers of English also get this wrong (prove vs. proof), sometimes spelling "prove" as "proove."
     
  10. Jul 8, 2017 #9

    SammyS

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    Figure from link in OP:
    upload_2017-7-8_11-40-23.png
     
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