# Uniform Convergence

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## uniform convergence

[′yü·nə‚fȯrm kən′vər·jəns]_{n }(

*x*)} converges uniformly on

*E*to ƒ(

*x*) if given ε > 0 there is an

*N*such that |ƒ

_{n }(

*x*) - ƒ(

*x*)| < ε="" for="" all="">

*x*in

*E*provided

*n*>

*N*.

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Uniform Convergence

an important special case of convergence. A sequence of functions *f _{n}* (

*x*) (

*n*= 1, 2, 3,…) is said to converge uniformly on a given set to the limit function

*f(x)*if, for every ∊ > 0, there exists a number

*N = N*(∊) such that, when η >

*N*, ǀ

*f*(

*x*) –

*f*

_{n}(

*x*)ǀ < ∊ for all points

*x*in the set.

For example, the sequence of functions *f*_{n}(*x*) = *x ^{n}* converges uniformly on the closed interval [0, 1 /2] to the limit

*f(x*) = 0. To show that this is true, let us take

*n*> In (1/∊)/In 2. It then follows that ǀ

*f(x*)–

*f*

_{n}(

*x*)ǀ ≤ (1/2)

^{n}< ∊ for all

*x*, 0 ≤ x ≤ 1/2. This sequence of functions, however, does not converge uniformly on the interval [0, 1]. Here, the limit function is

*f(x)*= 0 for 0 ≤

*x*< 1 and

*f(1) =*1. The reason for the failure to converge uniformly is that for arbitarily large

*η*there exist points η that satisfy the inequalities and for which ǀ

*f*(η) -

*f*

_{n}(

*η*)ǀ = η

^{n}> 1/2.

The notion of uniform convergence admits of a simple geometric interpretation. The uniform convergence of a sequence of functions *f*_{n}(*x*) on some closed interval to the function *f(x)* means that, for any ∊ > 0, all curves *y = f _{n}* (

*x*) with large enough

*n*will be located within a strip that is 2∊ in width and is bounded by the curves

*y*=

*f(x)*± ∊ for any

*x*in the interval (see Figure 1).

Uniformly converging sequences of functions have a number of important properties. For example, the limit of a uniformly converging sequence of continuous functions is also continuous. On the other hand, the example given above shows that the limit of a sequence of functions that does not converge uniformly may be discontinuous. An important role is played in mathematical analysis by Weierstrass’ theorem, which states that every function continuous on a closed interval can be represented as the limit of a uniformly converging sequence of polynomials. (*See*APPROXIMATION AND INTERPOLATION OF FUNCTIONS.)