Weierstrass function

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The Weierstrass function may also refer to the Weierstrass elliptic function ([\wp]) or the Weierstrass sigma, zeta or eta functions.

In mathematics, the Weierstrass function is a pathological example of a real-valued function on the real line. The function has the property that it is continuous everywhere but differentiable nowhere. In Weierstrass' original paper, the functions was defined by

[f(x)=\sum_^\infty a^n\cos(b^n\pi x),]

where [0

[ ab>1+\frac\pi.]

This construction along with the proof that it is nowhere differentiable was first given by Karl Weierstrass in a paper presented to the 'Konigliche Akademie des Wissenschaften' on the 18 of July 1872.

The proof that this function is continuous everywhere is elementary. Since the terms of the infinite series which defines it are bounded by [\pm a^n]and this has finite sum for [0 To prove that it is nowhere differentiable, we consider an arbitrary point [x \in ] and show that the function is not differentiable at that point. To do this, we construct two sequences of points [x_n] and [x'_n] which both converge to x, having the property that

[\lim \inf \frac > \lim \sup \frac.]

Naively it might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be 'small' in some sense. According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiablility points is somethine other than a finite set of points. Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functions, whose set of non-differentiability points must be a Lebesgue null set. When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz and has other nice properties.

The Weierstrass function could perhaps be described as one of the very first 'fractals', although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone. Kenneth Falconer in his book 'The Geometry of Fractal Sets', observes that the Hausdorff dimension of the classical Weierstrass function is bounded above by [\log a/\log b + 2], (where a and b are the constants in the construction above) and is generally believed to be exactly that value, but that this had not been proved rigourously.

The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass' original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear 'zigzag' function. G.H. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions [0

References

B.R. Gelbaum and J.M.H. Olmstead, Counterexamples in Analysis, Holden Day Publisher (June 1964).
Karl Weierstrass, Uber continuirliche Functionen eines reellen Arguments, die fur keiner Werth des letzeren einen bestimmten Differentialquotienten besitzen, Collected works.
K. Falconer, The Geometry of Fractal Sets, Oxford (1984)

External links

[Weierstrass function on MathWorld]

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