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