Principles of Mathematics in Operations Research

11

Continuity

In this chapter, we will define the fundamental notions of limits and continuity

of functions and study the properties of continuous functions. We will discuss

these properties in more general context of a metric space. The concept of

compactness will be introduced. Next, we will focus on connectedness and

investigate the relationships between continuity and connectedness. Finally,

we will introduce concepts of monotone and inverse functions and prove a set

of Intermediate Value Theorems.

11.1 Introduction I

Definition 11.1.1 Let (X, dx), (Y, dy) be two metric spaces; E ^ 0, E C X.

Let f : E i-> Y,p £ E,q € Y. We say limn^+p f(x) = q or f(x) —> q as x —» p

if Me > 0,3£ > 0 9 Vx G E with dx(x,p) < 5 we have dY(f{x), q) < e

(Le. Ve > 0,38 > 0 3 f{E D Bf(p)) C B»(g);.

Fig. 11.1. Limit and continuity

Definition 11.1.2 Let (X,dx), (Y,dY) be metric spaces; 0 ^ E C X, and
f : X t-¥ Y,p E E. f is said to be continuous at p if