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Analysis Question

  1. Mar 22, 2008 #1
    [SOLVED] Analysis Question

    1. The problem statement, all variables and given/known data
    Let f:Reals to Reals be a continuous at x=a, and further suppose f(a)>0. Show there exists an interval I about x=a such that f(x)>0 for all x in I.

    2. Relevant equations

    none

    3. The attempt at a solution
    i know the defn of continuity but I am not sure how to show this interval exists.
    I need HELP!
     
  2. jcsd
  3. Mar 22, 2008 #2
    well, this is quite easy to show, just use the definition of the limit, and chose epsylon in such a manner that it will allow you to do this. i.e [tex]0<\epsilon<f(a)-0[/tex].

    [tex]\forall\epsilon>0, \ \ also \ \ for 0<\epsilon<f(a),\exists\delta(\epsilon)>0[/tex] such that

    [tex]|f(x)-f(a)|<\epsilon, \ \ whenever \ \ 0<|x-a|<\delta[/tex] so from here we have:

    [tex] -\epsilon<f(x)-f(a)<\epsilon =>f(a)-\epsilon<f(x)<f(a)+\epsilon[/tex] but look now how we chose our epsilon, [tex]\epsilon<f(a)=>f(a)-\epsilon>0[/tex] so, eventually

    [tex]0<f(a)-\epsilon<f(x), \ \ whenever, \ \ \ xE(a-\delta,a+\delta[/tex].

    hope this helps.
     
    Last edited: Mar 22, 2008
  4. Mar 22, 2008 #3
    wow thanks
     
  5. Mar 24, 2008 #4
    Yeah, but make sure next time to show some work of yours!!.
     
  6. Mar 25, 2008 #5
    how do I know it is in the interval around a?
     
  7. Mar 25, 2008 #6
    But notice that x is from the interval [tex] (a-\delta,a+\delta)[/tex] which actually includes a. MOreover, it works only for that interval around a, for other intervals we are not sure.
     
    Last edited: Mar 25, 2008
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