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CONTINUITY in VS .NET
CONTINUITY Recognize QR Code 2d Barcode In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Creating QR Code JIS X 0510 In .NET Framework Using Barcode generation for .NET framework Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. [CHAP. 10
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10.7 Find the points of discontinuity (if any) of the functions whose graphs are shown in Fig. 108. Fig. 108 10.8 Give simple examples of functions such that: (a) f is de ned on [ 2, 2], continuous over [ 1, 1], but not continuous over [ 2, 2]. (b) g is de ned on [0, 1], continuous on the open interval (0, 1), but not continuous over [0, 1]. (c) h is continuous at all points except x = 0, where it is continuous on the right but not on the left. CHAP. 10] CONTINUITY
10.9 For each discontinuity of the following functions, determine whether it is a removable discontinuity. (a) The function f of Problem 7.4 (see Fig. 713). (b) The function f of example (c) in Section 8.3 [see Fig. 81(c)]. (c) The function f of example (a) in Section 9.1 [see Fig. 91(a)]. (d) The function f in example (b) in Section 9.2 (see Fig. 93). (e) The examples in Problems 10.3 and 10.4. 3x + 3 10.10 Let f be de ned by the formula f (x) = 2 . x 3x 4 (a) Find the arguments x at which f is discontinuous. (b) For each number a at which f is discontinuous, determine whether lim f (x) exists. If it exists, nd its value. (c) Write an equation for each vertical and horizontal asymptote of the graph of f . 10.11 Let f (x) = x + (1/x) for x = 0. (a) Find the points of discontinuity of f . (b) Determine all vertical and horizontal asymptotes of the graph of f . 10.12 For each of the following functions determine whether it is continuous over the given interval: 1 if x > 0 (a) f (x) = [x] over [1, 2] (b) f (x) = x over [0, 1] 0 if x = 0 (c) f (x) = 2x if 0 x 1 over [0, 1] x 1 if x > 1 if x = 4 if x = 4 if x = b if x = b (d) f as in part (c) over [1, 2] 2 x 16 10.13 If the function f (x) = x 4 c
is continuous, what is the value of c
2 x b2 10.14 Let b = 0 and let g be the function such that g(x) = x b 0 (a) Does g(b) exist (b) Does lim g(x) exist (c) Is g continuous at b
10.15 (a) Show that the following f is continuous: 4x + 1 3x + 7 f (x) = x 6 1
if x 1 and x = 6 4 if x = 6
algebra
( u v)( u + v) u v v= = u+ v u+ v
(b) For what value of k is the following a continuous function 7x + 2 6x + 4 f (x) = x 2 k if x 2 and x = 2 7 if x = 2 CONTINUITY
[CHAP. 10
10.16 Determine the points of discontinuity of the following function f : f (x) = 1 if x is rational 0 if x is irrational [Hint: A rational number is an ordinary fraction p/q, where p and q are integers. Recall Euclid s proof that 2 cannot be expressed in this form; it is an irrational number, as must be 2/n, for any integer n. It follows that any xed rational number r can be approached arbitrarily closely through irrational numbers of the form r + 2/n. Conversely, any xed irrational number can be approached arbitrarily closely through rational numbers.] 10.17 GC Use a graphing calculator to nd the discontinuities (if any) of the following functions: (a) f (x) = x+4 x + 4 (b) f (x) = x + 3 if x 2 if x > 2 x2 (c) f (x) = x2 1 x x (d) f (x) = 2 x 1 The Slope of a Tangent Line
The slope of a tangent line to a curve is familiar in the case of circles [see Fig. 111(a)]. At each point P of a circle, there is a line L such that the circle touches the line at P and lies on one side of the line (entirely on one side in the case of a circle). For the curve of Fig. 111(b), shown in dashed lines, L1 is the tangent line at P1 , L2 the tangent line at P2 , and L3 the tangent line at P3 . Let us develop a de nition that corresponds to these intuitive ideas about tangent lines. Fig. 111 Figure 112(a) shows the graph (in dashed lines) of a continuous function f . Remember that the graph consists of all points (x, y) such that y = f (x). Let P be a point of the graph having abscissa x. Then the coordinates of P are (x, f (x)). Take a point Q on the graph having abscissa x + h. Q will be close to P if and only if h is close to 0 (because f is a continuous function). Since the xcoordinate of Q is x + h, the ycoordinate of Q must be f (x + h). By the de nition of slope, the line PQ will have slope f (x + h) f (x) f (x + h) f (x) = (x + h) x h Observe in Fig. 112(b) what happens to the line PQ as Q moves along the graph toward P. Some of the positions of Q have been designated as Q1 , Q2 , Q3 , . . . , and the corresponding lines as M1 , M2 , M3 , . . . . These lines are getting

