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CONTINUITY
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[CHAP. 10
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2 x 4 (e) f (x) = x+2 4
10.7 Find the points of discontinuity (if any) of the functions whose graphs are shown in Fig. 10-8.
Fig. 10-8
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. 7-13). (b) The function f of example (c) in Section 8.3 [see Fig. 8-1(c)]. (c) The function f of example (a) in Section 9.1 [see Fig. 9-1(a)]. (d) The function f in example (b) in Section 9.2 (see Fig. 9-3). (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. 11-1(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. 11-1(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. 11-1 Figure 11-2(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 x-coordinate of Q is x + h, the y-coordinate 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. 11-2(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
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