Open AccessNote on the existence of converging sequences in certain oscillating successions of functions
Author(s) -
William Young
Publication year - 1916
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1916.0020
Subject(s) - countable set , neighbourhood (mathematics) , mathematics , pure mathematics , limit point , class (philosophy) , point (geometry) , function (biology) , closed set , object (grammar) , set (abstract data type) , limit (mathematics) , mathematical economics , discrete mathematics , mathematical analysis , computer science , geometry , artificial intelligence , evolutionary biology , biology , programming language
1. The object of the following note. is to prove that, in a large class of important cases, a succession of functions can be shown to contain sub-sets of functions which converge. The interest attaching to the question is well known, and is sufficiently iIlustrated by the use I have made of these considerations, for example, in my paper on the conditions that a trigonometrical series should have a certain form, as well as elsewhere. The first theorem on tho subject is, as is there pointed out, due to Arzelà. 2. Theorem 1.—Given a function which is upper semi-continuous on the left and lower semi-continuous on the right, there is a countable set of points dense everywhere, such that the value of the function at any point not belonging to the set is the unique limit of the values of the function at points of the set in a neighbourhood of the point when that neighbourhood shrinks up to the point .