By M. H. Protter C. B. Morrey Jr.
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Additional info for A First Course in Real Analysis
5 We denote by N x N the set of all ordered pairs of natural numbers (m, n). State and prove a Principle of mathematical induction for sets contained in NXN. 5 An asterisk is used to indicate difficult problems. 1 Continuity Most of the functions we study in elementary calculus are described by simple formulas. These functions almost always possess derivatives and, in fact, a portion of any first course in calculus is devoted to the development of routine methods for computing derivatives. However, not all functions possess derivatives everywhere.
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.
For example, letting f(x) = x 3 and g(x) = x 2 , we observe that f(x) < g(x) for -1 < x < 1, x =1= O. However, f(O) = g(O) = O. 10 (Sandwiching theorem). Suppose that f, g, and h are functions defined on the interval 0 < Ix - al < k for some positive number k. If f(x) < g(x) < h(x) on this interval, and if lim f(x) x ..... a = L, limh(x) x ..... a = L, then limx->ag(x) = L. PROOF. Given any € > 0, there are positive numbers c5, and c5 2 (which we may take smaller than k) so that 0 < Ix - al If(x) - LI <€ whenever Ih(x) - LI <€ whenever 0 < Ix and - < c5, < c52 • € < h(x) < L + € al In other words, L - € < f(x) < L + € and L for all x such that 0 < Ix - al < c5 where c5 is the smaller of c5, and c5 2 .
A First Course in Real Analysis by M. H. Protter C. B. Morrey Jr.