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A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators.
Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions or an improper integration of functions dependent on parameters. It is related to Abel's test for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts. The test is as follows.
In general, the most common criteria for pointwise convergence of a periodic function f are as follows: If f satisfies a Holder condition, then its Fourier series converges uniformly. [5] If f is of bounded variation, then its Fourier series converges everywhere. If f is additionally continuous, the convergence is uniform. [6]
The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. [2] The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.
If the domain of the functions is a topological space and the codomain is a uniform space, local uniform convergence (i.e. uniform convergence on a neighborhood of each point) and compact (uniform) convergence (i.e. uniform convergence on all compact subsets) may be defined. "Compact convergence" is always short for "compact uniform convergence ...
Note that () is continuous on the real closed interval [,] for <, by virtue of the uniform convergence of the series on compact subsets of the disk of convergence. Abel's theorem allows us to say more, namely that the restriction of G ( z ) {\displaystyle G(z)} to [ 0 , 1 ] {\displaystyle [0,1]} is continuous.
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.