Weierstrass sigma, zeta, or eta functions. Plot of Weierstrass function over the interval . In mathematics, the Weierstrass function is an example of a pathological real-valued function karl weierstrass fractals forex the real line.
Animation based on the increasing of the b value from 0. This construction, along with the proof that the function is not differentiable over any interval was first delivered by Weierstrass in a paper presented to the Königliche Akademie der Wissenschaften on 18 July 1872. Naïvely 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. The Weierstrass function could perhaps be described as one of the very first fractals studied, 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.
The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass’s original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear “zigzag” function. Wiener measure γ, the collection of functions that are differentiable at even a single point of has γ-measure zero. At least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes. University of Geneva, Switzerland, independently formulated a continuous, nowhere differentiable function that closely resembles Weierstrass’s function. Cellérier’s discovery was, however, published posthumously: Cellérier, C. The Hausdorff dimension of graphs of Weierstrass functions,” Proceedings of the American Mathematical Society, vol.