Sequence and Series of Functions with Applications to Continuity.

This paper focuses on sequences and series of functions and their connection with continuity. Basic concepts of convergence of sequences of functions are discussed, including pointwise convergence and uniform convergence. The differences between these types of convergence are explained using simple examples. Special attention is given to the continuity of functions and conditions under which the limit of a sequence of continuous functions remains continuous. The study also includes an introduction to series of functions and the continuity of their sum functions. The aim of this paper is to present the topic in a clear and elementary manner suitable for postgraduate students studying real analysis.