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Math Help - Need help on this proof

  1. #1
    Junior Member
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    Oct 2009
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    Need help on this proof

    I have to prove the following:

    Suppose that functions f and g are continuous at x=c\in (a,b) and f(c) > g(c). Prove there exists \delta > 0 such that for all x\in (a,b) with |x-c|<\delta, we have f(x) > g(x).

    I really have no idea. any help on this would be great thanks!
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  2. #2
    Newbie
    Joined
    Jun 2008
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    Hello

    First you can proof this result.

    If a,b)\longrightarrow R" alt="fa,b)\longrightarrow R" /> is a continuous function in c\in (a,b) then and f(c)>0, then there exists \delta>0 such that |x-c|<\delta, x\in (a,b) implies f(x)>0.

    This follows inmediately from continuous definition; taking \epsilon=f(x) there exists \delta>0 such that |x-c|<\delta, x\in (a,b) implies:

     |f(x)-f(c)|<f(c)\quad \Rightarrow\quad f(x)-f(c)>-f(c)\quad \Rightarrow\quad f(x)>0

    Then, for your exercise, apply this result to the funcion f-g. Note that the difference of continuous function is continuous.

    Best regards.
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  3. #3
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    [quote=el_manco;389581]taking \epsilon=f(x)
    do you mean \epsilon=f(c)??
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  4. #4
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    [quote=binkypoo;391521]
    Quote Originally Posted by el_manco View Post
    taking \epsilon=f(x)
    do you mean \epsilon=f(c)??
    Yes, of course. I meant \epsilon=f(c). Sorry for the typo.

    Best regards.
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