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Math Help - Continuity Problem

  1. #1
    Newbie Sterwine's Avatar
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    Continuity Problem

    Define h: R-> R by

    h(x) = { x^2, x is rational
    { 0, x is irrational

    Prove that h is not continous at c not equal to 0. Split the proof into two cases: one for c rational, the other for c irrational.
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by Sterwine View Post
    Define h: R-> R by

    h(x) = { x^2, x is rational
    { 0, x is irrational

    Prove that h is not continous at c not equal to 0. Split the proof into two cases: one for c rational, the other for c irrational.
    Well if c\in\mathbb{Q}, then \forall\delta>0, \exists x s.t. |c-x|<\delta and |f(c)-f(x)|=c^2. So given a point c, letting \epsilon=\frac{c^2}{2}>0 (for instance), will ensure that |c-x|<\delta and |f(c)-f(x)|\geq\epsilon.

    You can do a very similar thing for c irrational.
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