Van der Waerden's Continuous Nowhere Differentiable Function
About: This article is published in American Mathematical Monthly.The article was published on 1982-11-01. It has received 51 citations till now. The article focuses on the topics: Van der Waerden's theorem & Differentiable function.
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TL;DR: The history of the Takagi function and its application in various areas of mathematics, including number theory, combinatorics, and classical real analysis, is described in this article, where the authors present a survey of known properties of T including its nowhere-differentiability, modulus of continuity, graphical properties and level sets.
Abstract: This paper sketches the history of the Takagi function T and surveys known properties of T, including its nowhere-differentiability, modulus of continuity, graphical properties and level sets. Several generalizations of the Takagi function, in as far as they are based on the tent map, are also discussed. The final section reviews a number of applications of the Takagi function to various areas of mathematics, including number theory, combinatorics and classical real analysis.
103 citations
TL;DR: In this paper, the authors introduce the concept of metric-preserving functions, and present an introduction to metric preserving functions in the context of metric preservation functions, including the following:
Abstract: (1999). Introduction to Metric-Preserving Functions. The American Mathematical Monthly: Vol. 106, No. 4, pp. 309-323.
90 citations
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TL;DR: The Takagi function (x) is a continuous non-dierentiable function introduced by Teiji Takagi in 1903 that has appeared in a surprising number of dierent mathematical contexts, including mathematical analysis, probability theory and number theory.
Abstract: The Takagi function (x) is a continuous non-dierentiable function introduced by Teiji Takagi in 1903. It has appeared in a surprising number of dierent mathematical contexts, including mathematical analysis, probability theory and number theory. This paper surveys properties of this function.
72 citations
Posted Content•
TL;DR: In this article, a sufficient condition on the type space for revenue equivalence when the set of social alternatives consists of probability distributions over a finite set is given. But this condition is stronger than connectedness but weaker than smooth arcwise connectedness.
Abstract: We give a sufficient condition on the type space for revenue equivalence when the set of social alternatives consists of probability distributions over a finite set. Types are identified with real-valued functions that assign valuations to elements of this finite set, and the type space is equipped with the Euclidean topology. Our sufficient condition is stronger than connectedness but weaker than smooth arcwise connectedness. Our result generalizes all existing revenue equivalence theorems when the set of social alternatives consists of probability distributions over a finite set. When the set of social alternatives is finite, we provide a necessary and sufficient condition. This condition is similar to, but slightly weaker than, connectedness.
51 citations
Cites background from "Van der Waerden's Continuous Nowher..."
...…not boundedly gridwise connected between any pair of points s , s ′ ∈ S′; the graphs of continuous but nowhere differentiable functions (see, for instance, Billingsley 1982), and some straightforward modifications thereof, are the only examples of connected spaces with this property of which we…...
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...Arcwise connectedness (called also, by some authors, pathwise connectedness) does not imply bounded gridwise connectedness; for example, the graphs of some continuous but nowhere differentiable functions (see, for instance, Billingsley 1982) have the former but not the latter property....
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01 Jan 2003
TL;DR: In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points and even tried to give a theoretical justificati... as mentioned in this paper.
Abstract: In the early nineteenth century, most mathematicians believed that a continuous function has derivative at a significant set of points. A.~M.~Amp\`ere even tried to give a theoretical justificati ...
40 citations