Uniform convergence

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In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x.

The concept is important because several properties of the functions fn, such as continuity and Riemann integrability, are transferred to the limit f if the convergence is uniform.

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History

Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence.

The concept of uniform convergence was probably first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he used the phrase "convergence in a uniform way" when the "mode of convergence" of a series \textstyle{\sum_{n=1}^\infty f_n(x,\phi,\psi)} is independent of the variables φ and ψ. While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs.[1]

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