Uniform convergence
Uniform and point wise convergence for sequences of functions [ edit ] Main article: Uniform convergence In addition to sequences of numbers, one may also speak of sequences of functions on � ⊂ � , that is, infinite, ordered families of functions � � : � → � , denoted ( � � ) � = 1 ∞ , and their convergence properties. However, in the case of sequences of functions, there are two kinds of convergence, known as pointwise convergence and uniform convergence , that need to be distinguished. Roughly speaking, pointwise convergence of functions � � to a limiting function � : � → � , denoted � � → � , simply means that given any � ∈ � , � � ( � ) → � ( � ) as � → ∞ . In contrast, uniform convergence is a stronger type of convergence, in the sense that a uniformly convergent sequence of functions also converges pointwise, but not conversely. Uniform convergence requires members of the family of functions, � � , to fall within some error � > 0 of � for every value