Limits and Continuity of Functions

Abstract

In this chapter, we show the equivalency of two different definitions of limit, Cauchy’s and Heine’s. Then we classify the different kinds of discontinuity points which functions may possess. Moreover, we clarify the distinction between the concepts of continuous and uniformly continuous. Then we look at the main properties of sequences of functions and infinite series of functions: in particular, that the limit of an uniformly covergent sequence of continuous functions is still continuous, which might not happen for a sequence of continuous functions which is merely pointwise convergent. We lay out the hypotheses necessary in order to prove that a sequence of functions defined on a closed interval has an uniformly convergent subsequence (Arzela’s theorem). Results such as Cauchy’s criterion, the usual comparison tests, and Arzela’s theorem, are valid for complex-valued functions as well.