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Math Help - Continuity/Differentiability

  1. #1
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    Continuity/Differentiability

    Define a function h : \mathbb{R} \rightarrow \mathbb{R} by:

    (THIS IS A PIECEWISE FUNCTION):

    h(x) = {0 if x \in \mathbb{Q} .. x^3 + 3x^2 if x \not\in \mathbb{Q}

    1.) Determine which points h is continuous at, and determine which points h is discontinuous at. Prove these results.

    2.) Determine which points h is differentiable at, and determine which points h is non-differentiable at. Prove these results.
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    Quote Originally Posted by fifthrapiers View Post
    Define a function h : \mathbb{R} \rightarrow \mathbb{R} by:

    (THIS IS A PIECEWISE FUNCTION):

    h(x) = {0 if x \in \mathbb{Q} .. x^3 + 3x^2 if x \not\in \mathbb{Q}

    1.) Determine which points h is continuous at, and determine which points h is discontinuous at. Prove these results.

    2.) Determine which points h is differentiable at, and determine which points h is non-differentiable at. Prove these results.
    The function is continous when 0 = x^3+3x^2. Because when you approach a point x by rationals your limit is 0, when by irrationals the limit is x^3+3x^2. So need them to be equal for continuity to hold.
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    Quote Originally Posted by ThePerfectHacker View Post
    The function is continous when 0 = x^3+3x^2. Because when you approach a point x by rationals your limit is 0, when by irrationals the limit is x^3+3x^2. So need them to be equal for continuity to hold.
    Could you give me a proof for this? If it's continuous there, then it's discont. everywhere else? What about where it's diff/non-diff'able?

    Thanks TPH
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