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Math Help - analysis of derivatives

  1. #1
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    analysis of derivatives

    1) Let f(x)=x^2 sin(1/x^2) for x not equal to 0 and f(0)=0.
    a)Show a is differentiable on R
    b)Show that f ' is not bounded on the interval [-1,1].
    2) Let f(x)=x^2 if x is rational and f(x)=0 if x is irrational.
    a)Prove that f is continuous at exactly one point, namely at x=0
    b)Prove that f is differentiable at exactly one point, namely at x=0.
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  2. #2
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    For 2:
    Consider the sequence \{x_n\} such that x_n\to x .
    If x\neq{0} then f(x_n)\not\to f(x). This is because if x is rational take each x_n to be irrational, and vice-versa if x is irrational.

    For the second part, since f is only continuous at 0, this is the only point that it can possibly be differentiable. Now \displaystyle\frac{f(x_n)}{x_n} equals 0 is x_n is irrational and 0 if x_n is rational. So from this we see that f'(0)=0.
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