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(b) Assume that f (x) 0 for arguments to the right of and near a. Prove:
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(c) Formulate and prove results similar to parts (a) and (b) for lim f (x).
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9.11 GC Find the vertical and horizontal asymptotes of the graphs of the following functions with the help of a graphing calculator, and then use analytic methods to verify your answers. (a) 5 3x 3 4x 3 + x 1 x + 16 4 (d) x (b) (e) 4x 2 + 1 5 2x (c) |4 x| 4 x
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Continuity
10.1 DEFINITION AND PROPERTIES A function is intuitively thought of as being continuous when its graph has no gaps or jumps. This can be made precise in the following way. De nition: A function f is said to be continuous at a if the following three conditions hold: (i) lim f (x) exists.
(ii) f (a) is de ned. (iii) lim f (x) = f (a).
EXAMPLES
(a) Let f (x) = 0 if x = 0 . The function is discontinuous (that is, not continuous) at 0. Condition (i) is satis ed: lim f (x) = 1. x 0 1 if x = 0 Condition (ii) is satis ed: f (0) = 0. However, condition (iii) fails: 1 = 0. There is a gap in the graph of f (see Fig. 10-1) at the point (0, 1). The function is continuous at every point different from 0.
x a x a
(b) Let f (x) = x 2 for all x. This function is continuous at every a, since lim f (x) = lim x 2 = a 2 = f (a). Notice that there are no gaps or jumps in the graph of f (see Fig. 10-2).
Fig. 10-1
Fig. 10-2
Copyright 2008, 1997, 1985 by The McGraw-Hill Companies, Inc. Click here for terms of use.
CHAP. 10]
CONTINUITY
(c) The function f such that f (x) = [x] for all x is discontinuous at each integer, because condition (i) is not satis ed (see Fig. 7-12). The discontinuities show up as jumps in the graph of the function. (d) The function f such that f (x) = |x|/x for all x = 0 is discontinuous at 0 [see Fig. 8-1(b)]. lim f (x) does not exist and f (0) is not de ned. Notice that there is a jump in the graph at x = 0.
If a function f is not continuous at a, then f is said to have a removable discontinuity at a if a suitable change in the de nition of f at a can make the resulting function continuous at a. EXAMPLES In example (a) above, the discontinuity at x = 0 is removable, since if we rede ned f so that f (0) = 1, then the resulting function would be continuous at x = 0. The discontinuities of the functions in examples (c) and (d) above are not removable. A discontinuity of a function f at a is removable if and only if lim f (x) exists. In that case, the value of the
function at a can be changed to lim f (x).
A function f is said to be continuous on a set A if f is continuous at each element of A. If f is continuous at every number of its domain, then we simply say that f is continuous or that f is a continuous function. EXAMPLES
(a) Every polynomial function is a continuous function. This follows from example (b) of Property V in Section 8.2. (b) Every rational function h(x) = f (x) , where f and g are polynomials, is continuous at every real number a except the real roots g(x) (if any) of g(x). This follows from Property VI in Section 8.2.
There are certain properties of continuity that follow directly from the standard properties of limits (Section 8.2). Assume f and g are continuous at a. Then, (1) The sum f + g and the difference f g are continuous at a.
notation f + g is a function such that ( f + g)(x) = f (x) + g(x) for every x that is in both the domain of f and the domain of g. Similarly, ( f g)(x) = f (x) g(x) for all x common to the domains of f and g.
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