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

ISBN-10: 1461599903

ISBN-13: 9781461599906

ISBN-10: 146159992X

ISBN-13: 9781461599920

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**Additional resources for A First Course in Real Analysis**

**Example text**

Then lim f [g(x)] x~a = f(L). The proof is left to the reader. Remarks. 17 in which 00 may be replaced by + 00 or - 00. Also, one-sided limits may be used so that x ~ a + or a -. Additional theorems may be stated with L replaced by 00, + 00, or - 00. In all these theorems care must be used to ensure that the composite functionf[g] is actually defined on an appropriate domain. 17 illustrates this point. 17(a) Suppose that lim f(x) = x~a 00 and lim g(x) = a. x~-oo Suppose that for some positive number M, we have g( x) =1= a for x < - M.

A function f from a domain D in IR) to IR) is continuous at every isolated point of D. To see this let a be an isolated point of D and suppose I is such that {a} = InD. 3: One-sided limits-continuity on sets implies that x is in I. Then condition (i) in the definition of continuity is always satisfied since (i) holds when x = a, the only point in question. 2 carryover almost without change to the case of one-sided limits. 7, the limit of a composite function. 11. Suppose that f and g are functions on ~l to ~l.

20. 10 (Sandwiching theorem) for the case when x ~ - 00 (instead of x ~ a). 21. 18.

### A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.

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