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Thread: Show strictly monotone

  1. #1
    Super Member
    Mar 2006

    Show strictly monotone

    Show that if the function f is continuous on an interval and assumes no values more than once, then it is strictly monotone there.

    My proof is in the attachment, it is not finish, but would you please check to see if I have the right idea?

    Thank you.
    Attached Thumbnails Attached Thumbnails Show strictly monotone-problem-2.jpg  
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  2. #2
    Global Moderator

    Nov 2005
    New York City
    I did not really understand the attachment. Here is the idea of what you can to show. Let $\displaystyle I$ be an interval, say $\displaystyle [a,b]$ for simplicity sake (it can in fact be any interval, open, even unbounded but all the proofs are the same like here). And also it is given that $\displaystyle f$ is one-to-one and continous on $\displaystyle [a,b]$ (meaning it takes on a value exactly one time). We want to show it is strictly increasing (or decreasing). Pick any two points $\displaystyle x,y\in [a,b]$ with $\displaystyle x<y$. Say (without lose of generality) that $\displaystyle f(x) < f(y)$. Now argue that if $\displaystyle c<d$ then $\displaystyle f(c)<f(d)$ for all $\displaystyle c,d\in [a,b]$ by contradiction. And the contradiction is that if this was not the case then by the intermediate value theorem it is possible to find two points haviing the same value which is an impossibility since we are told the function is one-to-one.
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