Murray H. Protter, Charles Bradfield Morrey's A First Course in Real Analysis, Second Edition PDF

By Murray H. Protter, Charles Bradfield Morrey

ISBN-10: 0387974377

ISBN-13: 9780387974378

ISBN-10: 3540974377

ISBN-13: 9783540974376

Many adjustments were made during this moment variation of A First path in genuine research. the main seen is the addition of many difficulties and the inclusion of solutions to lots of the odd-numbered routines. The book's clarity has additionally been enhanced through the extra explanation of some of the proofs, extra explanatory comments, and clearer notation.

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Additional info for A First Course in Real Analysis, Second Edition

Example text

11 a > 0 <=>a is positive, a < 0 <=>a is negative, a> 0<=> -a< 0, a< 0<=> -a> 0, (v) if a and b are any numbers then exactly one of the three alternatives holds: a > b or a = b or a < b, (vi) a< b<=>b >a. (i) (ii) (iii) (iv) The next five theorems yield the standard rules for manipulating inequalities. We shall prove the first of these theorems and leave the proofs of the others to the reader. 12. If a, b, and c are any numbers and if a > b, then a+c>b+c and a-c>b-c. PROOF. ll(i) that a - b > 0.

As we progress in the study of analysis, it is important to enlarge substantially the class of functions under examination. Functions which possess derivatives everywhere form a rather restricted class; extending this class to functions which are differentiable except at a few isolated points does not enlarge it greatly. We wish to investigate significantly larger classes of functions, and to do so we introduce the notion of a continuous functions. Definitions. Suppose that f is a function from a domain Din IR 1 to IR 1• The function f is continuous at a if and only if (i) the point a is in an open interval I contained in D, and (ii) for each positive number 6 there is a positive number {)such that lf(x) - f(a)l < 6 whenever lx - al < b.

11. Suppose that f and g are functions on IR 1 to IR 1 . Iff is continuous at Land if g(x)-+ Las x-+ a+, then lim f[g(x)] = f(L). x-+a+ A similar statement holds if g(x)-+ Las x-+ a-. 7. Although we always require x > a when g(x)-+ L, it is not true that g(x) remains always larger than or always smaller than L. Hence it is necessary to assume that f is continuous at L and not merely continuous on one side. With the aid of the general definition of continuity it is possible to show that the function g: x -+ ~.

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A First Course in Real Analysis, Second Edition by Murray H. Protter, Charles Bradfield Morrey

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