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The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ( π x ) 2 n {\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}} takes the value 1 {\displaystyle 1} when x {\displaystyle x} is an integer and 0 {\displaystyle ...
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).
In Banach spaces, pointwise absolute convergence implies pointwise convergence, and normal convergence implies uniform convergence. For functions defined on a topological space, one can define (as above) local uniform convergence and compact (uniform) convergence in terms of the partial sums of the series.
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]
Pointwise Cauchy-convergence; Uniform Cauchy-convergence; Implications are cases of earlier ones, except: - Uniform convergence both pointwise convergence and uniform Cauchy-convergence. - Uniform Cauchy-convergence and pointwise convergence of a subsequence uniform convergence.
The distinction between pointwise and uniform convergence is important when exchanging the order of two limiting operations (e.g., taking a limit, a derivative, or integral) is desired: in order for the exchange to be well-behaved, many theorems of real analysis call for uniform convergence.
For uniform continuity, δ may depend on ε and ƒ. For pointwise equicontinuity, δ may depend on ε and x 0. For uniform equicontinuity, δ may depend only on ε. More generally, when X is a topological space, a set F of functions from X to Y is said to be equicontinuous at x if for every ε > 0, x has a neighborhood U x such that
Nevertheless, if the metric space M is complete, then any pointwise Cauchy sequence converges pointwise to a function from S to M. Similarly, any uniformly Cauchy sequence will tend uniformly to such a function. The uniform Cauchy property is frequently used when the S is not just a set, but a topological space, and M is a complete metric space ...