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Basic Proof Related to Continuity

  1. Mar 20, 2009 #1
    1. The problem statement, all variables and given/known data

    From Introduction to Topology by Bert Mendelson, Chapter 2.4, Exercise 8:

    Let R be the real numbers and f: R -> R a continuous function. Suppose that for some number [itex]a \in R[/itex], f(a) > 0. Prove that there is a positive number k and a closed interval [itex]F = [a - \delta, a + \delta][/itex] for some [itex]\delta > 0[/itex] such that [itex]f(x) \geq k[/itex] for [itex]x \in F[/itex].

    2. Relevant equations

    Neighborhood definition of continuity:

    Let f:(X, d) -> (Y, d'). f is continuous at a point [itex]a \in X[/itex] if and only if for each neighborhood M of f(a), [itex]f^{-1}(M)[/itex] is a neighborhood of a.

    3. The attempt at a solution

    I think my proof is right, I just want to make sure because it's a bit more involved than the proofs I've done up to this point:

    Let M be the closed ball about f(a) of radius f(a)/2. Because f is continuous, [itex]f^{-1}(M)[/itex] is a neighborhood of a. By the definition of neighborhood, there is an open ball about a of radius [itex]\eta[/itex]. All points in this ball map into M. Define a closed ball about a of radius [itex]\eta/2[/itex]. This ball is the range [itex][f-\delta,f + \delta][/itex] with [itex]\delta = \eta/2[/itex]. It will map exclusively into M, which has a lower bound of f(a)/2. Set k = f(a)/2.
     
  2. jcsd
  3. Mar 20, 2009 #2
    Youve got it! =)
     
  4. Mar 21, 2009 #3
    Thanks for looking it over!
     
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