By M. H. Protter C. B. Morrey Jr.
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Extra resources for A First Course in Real Analysis
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. For this purpose we need the concept of a one-sided limit. Definition.
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.
Then there is a unique sequence b p b 2 , . . , bk such that bl = a l and bi+l=bi+ai+1 for i=I,2, ... ,k-I if k>1. If k = 1, then b l = a l . 3) of the theorem, and we must show that this sequence is unique. Suppose b;, b;, ... , b~, b~+ I is a second sequence with the same properties. We have b; = bi for i = 1, 2, ... ,k because the assumption k E S implies these bi are unique. Now, b~+1 = b~ + ak + 1 = bk + ak +1 = bk + P and so (k + 1) E S. The set S is inductive and therefore S = N. The theorem is true for every natural number n.