A sequence of functions is said to uniformly converge to a function if for every positive real number , then there exists such that for all positive integers , we have . (More generally, we can replace with any metric space .)
This is different from pointwise convergence, where a sequence of functions converge pointwise if at every point in the domain, the functions converge. Uniform convergence is a stronger condition, because (speaking informally) the function has to converge at a similar rate everywhere on its domain.
If the functions continue to approach as shown, they converge uniformly.
Every uniformly convergent sequence converges pointwise, but the converse is not necessarily true. For example, the sequence of functions defined by for converges pointwise to the function , but this convergence is not uniform.
An equivalent definition is that if for all and
then converges uniformly.
Uniformly convergent sequences have a number of nice properties that pointwise convergent sequences do not necessarily have. A uniformly convergent sequence of continuous functions converges to a continuous function. A uniformly convergent sequence of differentiable functions defined on a closed interval converges to a differentiable function, and a sequence of Stieltjes-integrable functions converges to a Stieltjes-integrable function. It is possible to show by example that these properties do not have to hold for pointwise convergent functions.
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