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Math Help - Pathological example

  1. #1
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    Pathological example

    Will someone help me on this problem?
    Let f(x):R \rightarrow R be a continuous but differentiable nowhere ( for example the Weierstrass's function). Define F(x)=\int_0^x f(r)dr. Why does F(x) has the first derivative everywhere?
    Does F(x) have a second derivative?

    I think F(x) has the first derivative since that's why the FTOC part 2 says, but I'm stuck on part 2.
    Last edited by jackie; September 14th 2009 at 11:50 PM.
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  2. #2
    MHF Contributor chisigma's Avatar
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    By definition of 'integral function' if...

    F(x)= \int_{0}^{x} f(\tau)\cdot d\tau (1)

    ... then F(*) has prime derivative everywhere f(*) is continous and is...

    F^{'} (x)= f(x) (2)

    But f(*) in continous \forall x \in \mathbb{R} so that F^{'}(*) exists \forall x \in \mathbb{R}...

    Now from (2) we derive that...

    F^{''} (x) = f^{'} (x) (3)

    ... so that, because f^{'} (*) nowhere exists, the same is for F^{''} (*)...

    Kind regards

    \chi \sigma
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  3. #3
    MHF Contributor chisigma's Avatar
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    According to...

    Weierstrass Function -- from Wolfram MathWorld

    ... the 'Weierstrass function of degree a' is defined as...

    f_{a} (x) = \sum_{k=1}^{\infty} \frac {\sin \pi\cdot k^{a}\cdot x}{\pi\cdot k^{a}} (1)

    ... and it is everywhere continous but it's derivative exists only in a set of measure zero. More precisely in recent years it has been demostrated that the derivative exists and is f^{'}_{a} (x)= \frac{1}{2} only for x= \frac{2A+1}{2B+1}, A & B integers. The Weierstrass function written in the form (1) is an example of Fractal Fourier Series...

    Kind regards

    \chi \sigma
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