Pathological example

**jackie** · September 14th 2009, 11:29 PM

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.

**chisigma** · September 15th 2009, 12:28 AM

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

$\chi$ $\sigma$

**chisigma** · September 15th 2009, 01:44 AM

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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