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x 1 x 1 Recognize QR Code 2d Barcode In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Generate QRCode In VS .NET Using Barcode creation for VS .NET Control to generate, create Quick Response Code image in VS .NET applications. 10.2 Consider the function f such that f (x) = x [x] for all x. (See the graph of f in Fig. 96.) Find the points at which f is discontinuous. At those points, determine whether f is continuous on the right or continuous on the left (or neither). QR Code Scanner In .NET Using Barcode reader for .NET Control to read, scan read, scan image in VS .NET applications. Paint Bar Code In VS .NET Using Barcode creator for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. For each integer n, f (n) = n [n] = n n = 0. For n < x < n + 1, f (x) = x [x] = x n. Hence, Recognizing Bar Code In VS .NET Using Barcode decoder for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. QR Code Maker In Visual C# Using Barcode generation for .NET framework Control to generate, create Quick Response Code image in VS .NET applications. x n+
QR Code JIS X 0510 Creator In VS .NET Using Barcode generator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QR Maker In VB.NET Using Barcode maker for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in .NET applications. lim f (x) = lim (x n) = 0 = f (n) Bar Code Printer In .NET Framework Using Barcode encoder for .NET Control to generate, create bar code image in .NET applications. Matrix Barcode Drawer In .NET Using Barcode creation for .NET framework Control to generate, create Matrix Barcode image in .NET applications. x n+
Generate UPCA In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create UPCA image in .NET applications. Create USPS PLANET Barcode In .NET Framework Using Barcode maker for VS .NET Control to generate, create USPS Confirm Service Barcode image in VS .NET applications. Thus, f is continuous on the right at n. On the other hand, Code 128 Code Set C Creator In VS .NET Using Barcode drawer for Reporting Service Control to generate, create Code 128C image in Reporting Service applications. Create Code39 In None Using Barcode printer for Font Control to generate, create Code 3/9 image in Font applications. x n
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Barcode Generator In Java Using Barcode encoder for Java Control to generate, create barcode image in Java applications. UCC  12 Maker In VB.NET Using Barcode drawer for VS .NET Control to generate, create EAN128 image in VS .NET applications. so that f is not continuous on the left at n. It follows that f is discontinuous at each integer. On each open interval (n, n + 1), f coincides with the continuous function x n. Therefore, there are no points of discontinuity other than the integers. 10.3 For each function graphed in Fig.105, nd the points of discontinuity (if any). At each point of discontinuity, determine whether the function is continuous on the right or on the left (or neither). (a) There are no points of discontinuity (no breaks in the graph). (b) 0 is the only point of discontinuity. Continuity on the left holds at 0, since the value at 0 is the number approached by the values assumed to the left of 0. (c) 1 is the only point of discontinuity. At 1 the function is continuous neither on the left nor on the right, since neither the limit on the left nor the limit on the right equals f (1). (In fact, neither limit exists.) CONTINUITY
[CHAP. 10
(d) No points of discontinuity. (e) 0 and 1 are points of discontinuity. Continuity on the left holds at 0, but neither continuity on the left nor on the right holds at 1. Fig. 105 10.4 De ne f such that f (x) = (c) [0, 2] x for 0 x 1 . (See Fig. 106.) Is f continuous over: (a) [0, 1]; (b) [1, 2]; 2x 2 for 1 < x 2 (a) Yes, since f is continuous on the right at 0 and on the left at 1. (b) No, since f is not continuous on the right at 1. In fact, lim f (x) = lim (2x 2) = 0 = 1 = f (1) x 1+
x 1+
(c) No, since f is not continuous at x = 1, which is inside (0, 2). CHAP. 10] CONTINUITY
Fig. 106 10.5 For each of the following functions, determine the points of discontinuity (if any). For each point of discontinuity, determine whether it is removable. x2 1 x2 if x 0 for x = 1 (b) g(x) = (a) f (x) = x 1 x 2 if x < 0 (a) There are no points of discontinuity [see Fig. 107(a)]. At x = 0, f (0) = 0 and lim f (x) = 0. (b) The only discontinuity is at x = 1, since g(1) is not de ned [see Fig. 107(b)]. This discontinuity is removable. Since (x 1)(x + 1) x2 1 = = x + 1, lim g(x) = lim (x + 1) = 2. So, if we de ne the function value at x = 1 to be x 1 x 1 x 1 x 1 2, the extended function is continuous at x = 1. Fig. 107 Supplementary Problems
10.6 Determine the points at which each of the following functions is continuous. (Draw the graphs of the functions.) Determine whether the discontinuities are removable.

