Introduction to Mathematical Analysis I - 3rd Edition

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 definition 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 first introduce the notion of limit point of a set. Definition 3.1.1 Let Dbe a subset of R. A point x0 ∈R(not necessarily in D) is called a limit point of Dif for every δ >0, the open interval (x0 −δ,x0 +δ) contains a point x of D, x̸ =x0. 1 Definition 3.1.2 A point x0 ∈Dwhich is not a limit point of Dis called anisolated point of D. Note that a point x0 in Ris a limit point of a set Dif 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 Dbe a subset of R. The following are equivalent: (i) The point x0 is a limit point of D. (ii) There exists a sequence {xn}inDsuch that xn̸ =x0 for all n∈Nand limn→∞xn =x0. ■ Example 3.1.1 The following examples illustrate the definition of limit point. (a) Let D= [1,3). Then every point of Dis a limit point. Moreover, 3 is a limit point of Das well. The set Dhas 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 Ddoes not have any limit points. Every element of Nis an isolated point of N. 1Other authors refer to limit points as accumulation points or cluster points.

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