the limit of a uniformly convergent sequence of continuous functions is continuous
Proof. Let , where and are metric spaces. Suppose uniformly and each is continuous. Then given any , there exists such that implies for all . Pick an arbitrary larger than . Since is continuous, given any point , there exists such that implies . Therefore, given any and , there exists such that
Therefore, is continuous.
The theorem also generalizes to when is an arbitrary topological space. To generalize it to an arbitrary topological space, note that if for all , then , so is a neighbourhood of . Here denote the open ball of radius , centered at .
|Title||the limit of a uniformly convergent sequence of continuous functions is continuous|
|Date of creation||2013-03-22 15:21:58|
|Last modified on||2013-03-22 15:21:58|
|Last modified by||neapol1s (9480)|