1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook