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Math Help - differentiability of functions

  1. #1
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    differentiability of functions

    We had a problem in class today which read
    f(x) = x^2 if xЭQ
    f(x) = 0 otherwise.
    We are studying differentiability, and my tutor said that this function was differentiable at 0 but not at any other point.

    I can see how to show its differentiable at 0, but I dont know how I would go about proving that is isn't differentiable at any other point.
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  2. #2
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    Quote Originally Posted by nlews View Post
    We had a problem in class today which read
    f(x) = x^2 if xЭQ
    f(x) = 0 otherwise.
    We are studying differentiability, and my tutor said that this function was differentiable at 0 but not at any other point.

    I can see how to show its differentiable at 0, but I dont know how I would go about proving that is isn't differentiable at any other point.

    Take any point x_0\neq 0 , and choose now a sequence of rational points converging to it, and then a seq. of irrational points converging to it.

    If the limit \lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0} exists it must exist no matter how we choose to make x\to x_0 ...

    Tonio
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  3. #3
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    Im sorry but I don't really follow that.
    If i take two sequences both converging to the same point, how does that relate to the second part? Sorry to be a pain.
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  4. #4
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    The derivative, at x= a, of f(x) is defined as
    \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}

    Further, \lim_{x\to a} g(x)= L if and only if \lim_{n\to \infty} g(a_n)= L for every sequence \{a_n\} converging to a.

    Now, if there exist two sequences, \{a_n\} and \{b_n\}, converging to a, such that \lim_{n\to\infty}\frac{f(a_n)- f(a)}{a_n-a}\ne \lim_{n\to\infty}\frac{f(b_n)- f(a)}{b_n- a} then the function is NOT differentiable at a. That is, if the two sequences have different limits, then the limit as h goes to 0 cannot exist and so the function is differentiable. For any a other than 0, take \{a_n\} to be a sequence of rational numbers converging to a, and take \{b_n\} to be a sequence of irrational numbers converging to a.
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  5. #5
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    Quote Originally Posted by HallsofIvy View Post
    The derivative, at x= a, of f(x) is defined as
    \lim_{h\to 0}\frac{f(a+h)- f(a)}{h}

    Further, \lim_{x\to a} g(x)= L if and only if \lim_{n\to \infty} g(a_n)= L for every sequence \{a_n\} converging to a.

    Now, if there exist two sequences, \{a_n\} and \{b_n\}, converging to a, such that \lim_{n\to\infty}\frac{f(a_n)- f(a)}{a_n-a}\ne \lim_{n\to\infty}\frac{f(b_n)- f(a)}{b_n- a} then the function is NOT differentiable at a. That is, if the two sequences have different limits, then the limit as h goes to 0 cannot exist and so the function is differentiable. For any a other than 0, take \{a_n\} to be a sequence of rational numbers converging to a, and take \{b_n\} to be a sequence of irrational numbers converging to a.
    Why not using the same techniques show that f is not continuous at any x\ne 0? This is easier in this case and the non-differentiability follows immediately.
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