Chapter 6 - CONTINUITY

Abstract

This chapter provides an overview of continuity. The chapter discusses limits and continuity of functions. A function which is continuous at each point of its domain is said to be a continuous function. From many results presented in the chapter, it seems that a measurable function is obtained by any reasonable operations using measurable functions or sequences of measurable functions. The chapter reviews uniformity, Tietze extension theorem, and Weierstrass approximation theorem. A real-valued continuous function on an interval may be approximated uniformly by a polynomial. A polynomial is said to be a rational polynomial if all its coefficients are rational numbers. The chapter also reviews absolute continuity. A theory is presented that shows how intimately absolute continuity and Lebesgue integrability are related. It further discusses equicontinuity, and semicontinuity. Limits inferior and superior for sequences of sets and sequences of real numbers are extended to real-valued functions. All continuous functions with measurable domains are measurable functions. The chapter shows how nearly the converse comes to being true.