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10.2 Consider the function f such that f (x) = x [x] for all x. (See the graph of f in Fig. 9-6.) 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).
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For each integer n, f (n) = n [n] = n n = 0. For n < x < n + 1, f (x) = x [x] = x n. Hence,
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lim f (x) = lim (x n) = 0 = f (n)
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Thus, f is continuous on the right at n. On the other hand,
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lim f (x) = lim [x (n 1)] = n (n 1) = 1 = 0 = f (n)
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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.10-5, 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. 10-5
10.4 De ne f such that f (x) = (c) [0, 2]
x for 0 x 1 . (See Fig. 10-6.) 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. 10-6 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. 10-7(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. 10-7(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. 10-7
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.
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