Uniform convergence

as the function domain if, given any arbitrarily small positive number

The difference between uniform convergence and pointwise convergence was not fully appreciated early in the history of calculus, leading to instances of faulty reasoning.

The concept, which was first formalized by Karl Weierstrass, is important because several properties of the functions

, such as continuity, Riemann integrability, and, with additional hypotheses, differentiability, are transferred to the limit

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect.

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.

[2] Later Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent (German: uniformly convergent) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894.

Independently, similar concepts were articulated by Philipp Ludwig von Seidel[3] and George Gabriel Stokes.

G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis."

Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others.

We first define uniform convergence for real-valued functions, although the concept is readily generalized to functions mapping to metric spaces and, more generally, uniform spaces (see below).

is not quite standardized and different authors have used a variety of symbols, including (in roughly decreasing order of popularity): Frequently, no special symbol is used, and authors simply write to indicate that convergence is uniform.

is a complete metric space, the Cauchy criterion can be used to give an equivalent alternative formulation for uniform convergence:

Note that interchanging the order of quantifiers in the definition of uniform convergence by moving "for all

To make this difference explicit, in the case of uniform convergence,

One may straightforwardly extend the concept to functions E → M, where (M, d) is a metric space, by replacing

Uniform convergence admits a simplified definition in a hyperreal setting.

, a basic example of uniform convergence can be illustrated as follows: the sequence

(here the upper square brackets indicate rounding up, see ceiling function).

In this example one can easily see that pointwise convergence does not preserve differentiability or continuity.

The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset

The Weierstrass M-test requires us to find an upper bound

independent of the position in the disc: To do this, we notice and take

are topological spaces, then it makes sense to talk about the continuity of the functions

This theorem is an important one in the history of real and Fourier analysis, since many 18th century mathematicians had the intuitive understanding that a sequence of continuous functions always converges to a continuous function.

The uniform limit theorem shows that a stronger form of convergence, uniform convergence, is needed to ensure the preservation of continuity in the limit function.

For the Riemann integral, this can be done if uniform convergence is assumed: In fact, for a uniformly convergent family of bounded functions on an interval, the upper and lower Riemann integrals converge to the upper and lower Riemann integrals of the limit function.

converges: With this definition comes the following result: Let x0 be contained in the set E and each fn be continuous at x0.

In other words, almost uniform convergence means there are sets of arbitrarily small measure for which the sequence of functions converges uniformly on their complement.

"Uniform convergence", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

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).
Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions converge to the non-continuous red function. This can happen only if convergence is not uniform.