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Uniform Continuity

  1. Oct 24, 2010 #1
    1. The problem statement, all variables and given/known data
    a) Give an example of a bounded continuous function f: R -> R which is not uniformly continuous.

    b) State (in terms of a small Epsilon and a large K) what it means to say that f(x) -> 0 as x -> infinity (plus or minus)

    c) Now assume that f: R -> R is continuous and f(x) -> 0 as x -> infinity (plus or minus). Show that f is uniformly continuous.

    2. Relevant equations

    3. The attempt at a solution
    a) So we need a function that is bounded and continuous, but has an unbounded derivative.

    sin(x^2), because 2xsin(x^2) is not bounded. There is not a well-behaved delta for any two values in the domain. No matter how small an epsilon we pick, for x large enough, f(x) will range between -1, 1 for values (x-e, x+e).

    b) Since KE = delta, f(x) -> 0 as x -> infinity (+ or -) means that d(f(x) + E, f(x) - E) -> 0 as d(x + delta, x - delta) gets larger.

    c) Let E > 0, delta >0. Take x,y contained in R such that d(x,y) < delta. Since d(f(x), f(y)) -> 0 as x-> infinity (+ or -), d(f(x), f(y)) = 0 for large enough x. So we set delta = E and d(f(x) -f(y)) = 0 < delta = E.
  2. jcsd
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