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Homework Help: Continuity & Properties of functions

  1. May 7, 2008 #1
    1. The problem statement, all variables and given/known data

    Let f:R->R be a continuous function where limit as x goes to positive/negative infinity is negative infinity. Prove that f has a maximum value on R.

    2. Relevant equations
    None


    3. The attempt at a solution
    I tried to use the definition of infinite limits but I'm not sure how to do this explicitly.
     
  2. jcsd
  3. May 7, 2008 #2

    HallsofIvy

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    Science Advisor

    Let X be a real number. Since [itex]\lim_{x\rightarrow \infty}= -\infty[/itex], there exist x1[/sup] such that if x> x1, f(x)< X. Since [itex]\lim_{x\rightarrow -\infty}= -\infty[/itex], there exist x2 such that if x< x2, f(x)< X. Now look at the values of f(x) on the closed and bounded interval [x1, x2].
     
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