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Continuity of piecewise defined trig functions

  1. Sep 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Define functions f and g on [-1,1] by

    f(x) = xcos(1/x) if x≠0 and 0 if x = 0

    g(x)= cos(1/x) if x≠0 and 0 if x = 0

    (These are piecewise defined. I don't know how to type them in here.)

    Prove that f is continuous at 0 and that g is not continuous at 0. Explain why these functions are continuous at every other point in [-1,1].

    2. Relevant equations



    3. The attempt at a solution

    Using the definition of continuity,

    Suppose g is continuous at x=0.

    Since g is continuous for all x in [-1,1], then for all ε>0, there exists a δ > 0 such that |g(x)-g(c)| < ε for all x in [-1,1] that satisfy |x-c| < δ

    Let ε=1/2
    Let c = 0
    Choose δ=ε

    Then |g(x)-g(c)| = |cos(1/x)| and since -1 ≤ cos(1/x) ≤ 1 for all x in [-1,1], then |cos(1/x)|≤|2x|<2δ=2/2=1

    Then if x=3/[itex]\pi[/itex], |cos(1/x)| = |1/2| < 1/2 which 1/2 cannot be less than itself. So there exists a ε where the continuity definition fails at c=0 and thus g(x) is not continuous on [-1,1].

    Is this correct for showing g(x) is not continuous at x=0?

    I am unsure how to show f(x) is continuous at x=0 since it is a product of two functions where one of them is not continuous at 0.

    I have a theorem that will help me with showing that trig functions are continuous if they are defined on their domain.
     
  2. jcsd
  3. Sep 8, 2012 #2

    jbunniii

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    I didn't understand your argument for proving that g is not continuous at 0. I don't think it's right. Can you show that in any neighborhood around x = 0, g(x) takes on the values +1 and -1 for infinitely many x? Can you use that fact to show that g is not continuous at 0?

    To show that f is continuous at 0, try sandwiching it between two continuous functions which both have the same value at x = 0.
     
  4. Sep 8, 2012 #3

    HallsofIvy

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    Are you really required to do an "epsilon-delta" proof? It is relatively straight forward to find the limits of both f and g (or show they don't exist) as x goes to 0.
     
  5. Sep 8, 2012 #4
    Jbunnlll, if I understand what you are saying, I was trying to use that fact to make a contradiction using the definition of continuity.

    Hallsofivy, we aren't required to use the epsilon-delta definiton. It should work for any case, so I was really testing my knowledge on the definiton. I will try your suggestion. Sounds a lot easier!
     
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