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

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

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

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The function f is continuous on the left at a if and only if a is in the domain offandf(x)~f(a) as x~a-. If the domain of a function f is a finite interval, say a < x < b, then limits and continuity at the endpoints are of the one-sided variety. For example, the functionj: X~x2 - 3x + 5 defined on the interval 2 < x < 4 is continuous on the right at x = 2 and continuous on the left at x = 4. The following general definition of continuity for functions from a set in IR) to IR) declares that such a function is continuous on the closed interval 2 < x < 4.

1 f(x) *17. 18. o(sin(l / x)) does not exist. *19. ox loglxl = o. 20. The function f(x) = x cot x is not defined at x = O. Can the domain of f be enlarged to include x = 0 in such a way that the function is continuous on the enlarged domain? 2 Theorems on limits The basic theorems of calculus depend for their proofs on certain standard theorems on limits. These theorems are usually stated without proof in a first course in calculus. In this section we fill the gap by providing proofs of the customary theorems on limits.

Therefore S = N. D We now illustrate how the principle of mathematical induction is applied in practice. 22 as the statement of the familiar principle of mathematical induction. 22. EXAMPLE. 1. Show that 1+2 ... 1 ) for every natural number n. Solution. 1) holds. We shall show that S is an inductive set. 1) holds for n = 1. (b) Suppose k E S. 1) holds with n = k. Adding (k + 1) to both sides we see that 1+ ... +k+(k+ 1)= k(k+l) (k+l)(k+2) 2 +(k+ 1)= 2 which is the formula for n = k + 1. Thus (k + 1) is in S whenever k is.

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