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By M. H. Protter C. B. Morrey Jr.

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Extra info for A First Course in Real Analysis

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Sn is written Sl n S2 n ... n Sn or, briefly, n7=IS;, It may happen that two sets Sl and S2 have no elements in common. In such a case their intersection is empty, and we use the term empty set for the set which is devoid of members. 5 Most often we will deal with sets each of which is specified by some property or properties of its elements. For example, we may speak of the set of all even integers or the set of all rational numbers between and 1. We employ the special symbol ° {x: x=2n and is an integer} n to represent the set of all even integers.

11. 8. 12. Prove that if a * 0 and F: x ~ 1/ x, then F is continuous at a. 13. Prove that if a > 0 and G: x ~ Vx , then G is continuous at a. 3 One-sided limits-continuity on sets The function j: x ~ Yx is continuous for all x > 0 and, since YO = 0, it is clear that f(x)~ f(O) as x tends to 0 through positive values. 1 is not fulfilled at x = O. We now wish to extend the definition of continuous function so that a function such as Yx will have the natural property of continuity at the endpoint of its domain.

5), and S2 is the vertically shaded set, then Sl U S2 consists of the entire shaded area, and Sl n S2 consists of the doubly shaded area. Similarly, we may form the union and intersection of any number of sets. When we write Sl U S2 U ... U Sn for the union S of the n sets Sl' S2' ... , Sn' then S consists of all elements each of which is in at least one of the n sets. We also use the notation S = U 7= IS; as a shorthand for the union of n sets. The intersection of n sets Sp S2' ... , Sn is written Sl n S2 n ...

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