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Homework Help: Analysis help

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data

    If [itex]f:\mathbb{R} \to \mathbb{R}[/itex] is such that for all [itex]r>0[/itex] there exists a continuous function [itex]g_r \mathbb{R} \to \mathbb{R}[/itex] such that [itex]|g_r (x) - f(x)| < r[/itex] for [itex]|x| < 1[/itex] then f is continuous at 0.


    2. Relevant equations


    3. The attempt at a solution
    When [itex]|x| < \delta _g[/itex], [itex]|g_r (x) - g_r (0)| < \epsilon '[/itex] ...(1)
    When [itex]|x| < 1[/itex], [itex]|g_r (x) - f(x)| < r[/itex] , i.e. [itex]|f(x) - g_r (x)| < r[/itex] ...(2)
    [itex]|g_r(0) - f(0)| < r[/itex] ...(3)

    Adding (1) and (2) gives [itex]|f(x) - g_r (0)| < \epsilon ' + r[/itex] ...(4)
    Adding (3) and (4) gives [itex]|f(x) - f(0)| < \epsilon ' + 2r[/itex]
    So when |x| is the [itex] \min\{1, \delta_g\}[/itex] , [itex]|f(x) - f(0)| < \epsilon[/itex]
    And by the previous result, we can say that f is continuous at 0.

    Not sure whether its right for me to take the [itex]\delta = \min\{1, \delta_g\}[/itex]
     
  2. jcsd
  3. May 3, 2010 #2

    lanedance

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    Homework Helper

    I think it is ok, though it doesn't make much sense to have delta greater than 1 anyway...
     
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