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(2) If c is a constant, the function cf is continuous at a. in .NET framework
(2) If c is a constant, the function cf is continuous at a. QR Decoder In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. Denso QR Bar Code Printer In .NET Framework Using Barcode drawer for .NET framework Control to generate, create QR Code image in Visual Studio .NET applications. notation cf is a function such that (cf )(x) = c f (x) for every x in the domain of f .
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the left at a if it satis es the conditions for continuity at a with lim replaced by lim . Note that f is continuous at a if x a x a
and only if f is continuous both on the right and on the left at a, since lim f (x) exists if and only if both lim f (x) and lim f (x) exist and lim f (x) = lim f (x). x a x a+ x a x a x a +
EXAMPLES
(a) The function f (x) = x + 1 if x 1 is continuous on the right at 1 (see Fig. 103). Note that lim f (x) = lim (x + 1) = x if x < 1 x 1+ x 1+ 2 = f (1). On the other hand, f is not continuous on the left at 1, since lim f (x) = lim x = 1 = f (1). Consequently, f is not continuous at 1, as is evidenced by the jump in its graph at x = 1. x 1 x 1
(b) The function of example (c) of Section 8.3 [see Fig. 81(c)] is continuous on the left, but not on the right at 1. (c) The function of example (b) of Section 9.1 [see Fig. 91(b)] is continuous on the right, but not on the left at 1. (d) The function of example (b) of Section 9.2 (see Fig. 93) is continuous on the left, but not on the right at 0. Fig. 103 10.3 CONTINUITY OVER A CLOSED INTERVAL
We shall often want to restrict our attention to a closed interval [a, b] of the domain of a function, ignoring the function s behavior at any other points at which it may be de ned. De nition: A function f is continuous over [a, b] if: (i) f is continuous at each point of the open interval (a, b). (ii) f is continuous on the right at a. (iii) f is continuous on the left at b. EXAMPLES (a) Figure 104(a) shows the graph of a function that is continuous over [a, b]. (b) The function f (x) = 2x 1 if 0 x 1 is continuous over [0, 1] [see Fig. 104(b)]. otherwise CHAP. 10] CONTINUITY
Fig. 104 Note that f is not continuous at the points x = 0 and x = 1. Observe also that if we rede ned f so that f (1) = 1, then the new function would not be continuous over [0, 1], since it would not be continuous on the left at x = 1. Solved Problems
2 x 1 10.1 Find the points at which the function f (x) = x + 1 2 if x = 1 if x = 1 is continuous. For x = 1, f is continuous, since f is the quotient of two continuous functions with nonzero denominator. Moreover, for x = 1, x2 1 (x 1)(x + 1) = =x 1 x+1 x+1 whence lim f (x) = lim (x 1) = 2 = f ( 1). Thus, f is also continuous at x = 1. f (x) =

