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### Weierstrass Curve

```Date: 09/13/2002 at 01:11:24
From: Colin
Subject: Weierstrass curve

ago the mathematician Weierstrass gave an example of a curve
consisting of angles, or corners, and nothing else. Where can I find
this equation?
```

```
Date: 09/13/2002 at 10:26:59
From: Doctor Fenton
Subject: Re: Weierstrass curve

Hi Colin,

Weierstrass's original example was a trigonometric series, and
checking the literature, I found several different versions of it, so
I'm not sure exactly which version was Weierstrass's.

One version (Gelbaum and Olmstead, _Counterexamples in Analysis_) is

oo
---
\    b^n cos(a^n pi x)
/
---
n=0

where b is an odd integer, 0<a<1, and ab > 1+(3*pi/2).

T. W. Körner, in _Fourier Analysis_, gives the example as

oo
---
\    a^(-n) sin(b^n x)
/
---
n=-oo

where b is an integer, and b/a and a are "sufficiently large."

Neither book proves the properties of the function; it appears to be
pretty complicated to analyze.  Instead, many books prove the result
for similar but simpler functions.

Walter Rudin uses a function g(x)=|x| for |x| < 1, and extends g(x) to
the real line by making it periodic of period 2.  He proves that

oo
---
\    (3/4)^n g(4^n x)
/
---
n=0

is continuous and nowhere differentiable. Michael Spivak has a similar
example in his text _Calculus_.

Körner also proves that

oo
---
\    (n!)^(-1) sin((n!)^2 x)
/
---
n=0

is continuous and nowhere differentiable.

I did a quick Web search using keywords "Weierstrass" and
"nondifferentiable" and found many references, but no explicit
formulas, but then I didn't check many sites. You might also use
"nowhere differentiable" instead of nondifferentiable if you want
to try that search.

If you have any questions, please write back.

- Doctor Fenton, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Analysis
College Calculus
High School Analysis
High School Calculus

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