By Murray H. Protter, Charles Bradfield Morrey

Many adjustments were made during this moment version of A First path in actual research. the main visible is the addition of many difficulties and the inclusion of solutions to lots of the odd-numbered workouts. The book's clarity has additionally been greater by means of the additional rationalization of a number of the proofs, extra explanatory feedback, and clearer notation.

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**Extra resources for A First Course in Real Analysis, Second Edition **

**Sample text**

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 function f: x -+ x 2 - 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 ~ 1 to ~ 1 declares that such a function is continuous on the closed interval 2 ~ x ~ 4. Definitions. Let f be a function from a domain D in ~ 1 to ~ 1 • The function f is continuous at a with respect to D if and only if (i) a is in D, and (ii) for each e > 0 there is a {> > 0 such that lf(x) - f(a)l < e whenever xeD and Ix - al < f>.

If a 1 , a 2 , ••• , am, am+l• ... , m+n L i=l m+n m a; = am+n is any sequence, then L a; L a; + i=m+l i=l and Da; = Da; m+n ( m ( ) · m+n i=Ul ) a; · The extended associative law, the extended commutative law, and the extended distributive law with which the reader is familiar from elementary algebra, are stated in the next three propositions. 5. The sum of a finite sequence can be obtained by separating the given sequence into several shorter sequences, adding the terms in each of these, and then adding the results.

2 used the inequality 0 < Ix - al < f>. As a result, all the theorems on limits imply corresponding theorems for limits from the left and limits from the right. The definition of one-sided limits leads to the definition of one-sided continuity. Definitions. A function f is continuous on the right at a if and only if a is in the domain off and f(x)-+ f(a) as x -+ a+. The function f is continuous on the left at a if and only if a is in the domain off and f(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.