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Uniform Convergence

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Book cover Essential Real Analysis

Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

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Abstract

This chapter focuses on uniform convergence and provides the building blocks and basic results needed for the construction of continuous, smooth and analytic functions using infinite series-the subject of chapter “Functions”. The chapter concludes with van der Waerden’s example of a continuous nowhere differentiable function.

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Notes

  1. 1.

    More than 100 years later Gerver showed in 1969 that ∑ n = 1 ∞sin(n 2 x)∕n 2 is differentiable iff x = pπ∕q, where p, q are odd integers.

  2. 2.

    In 1916, G.H. Hardy improved this result to ab > 1.

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Authors and Affiliations

  1. Engineering Mathematics Department, Merchant Venturers School of Engineering, Bristol University, Bristol, UK

    Michael Field

  2. Department of Mathematics, Rice University, Houston, Texas, USA

    Michael Field

Authors
  1. Michael Field
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Field, M. (2017). Uniform Convergence. In: Essential Real Analysis. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-67546-6_4

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