Limits of Functions Limit Theorems Continuity Properties of Continuous Functions Uniform Continuity 3. LIMITS AND CONTINUITY In this chapter, we extend our analysis of limit processes to functions and give the precise defnition of continuous function. We derive rigorously two fundamental theorems about continuous functions: the extreme value theorem and the intermediate value theorem. 3.1 Limits of Functions We frst introduce the notion of limit point of a set. Defnition 3.1.1 Let D be a subset of R. A point x0 ∈ R (not necessarily in D) is called a limit point of D if for every δ > 0, the open interval (x0 − δ ,x0 + δ ) contains a point x of D, x ̸ = x0. 1 Defnition 3.1.2 A point x0 ∈ D which is not a limit point of D is called an isolated point of D. Note that a point x0 in R is a limit point of a set D if it can be approximated arbitrarily close by elements of D. The following proposition makes this statement precise. The proof is left as an exercise. Proposition 3.1.1 Let D be a subset of R. The following are equivalent: (i) The point x0 is a limit point of D. (ii) There exists a sequence {xn} in D such that xn ̸ = x0 for all n ∈ N and limn →∞ xn = x0. ■ Example 3.1.1 The following examples illustrate the defnition of limit point. (a) Let D =[1,3). Then every point of D is a limit point. Moreover, 3 is a limit point of D as well. The set D has no isolated points. In general, if an interval has more than one point, then every point of the interval is a limit point. If in addition, the interval is bounded, its endpoints are limit points as well. (b) Let D = N. Then D does not have any limit points. Every element of N is an isolated point of N. 1Other authors refer to limit points as accumulation points or cluster points.
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