(2) If c is a constant, the function cf is continuous at a. in .NET framework

Generation QR Code in .NET framework (2) If c is a constant, the function cf is continuous at a.

(2) If c is a constant, the function cf is continuous at a.
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notation cf is a function such that (cf )(x) = c f (x) for every x in the domain of f .
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(3) The product fg is continuous at a, and the quotient f/g is continuous at a provided that g(a) = 0.
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notation fg is a function such that ( fg)(x) = f (x) g(x) for every x that is in both the domain of f and the domain of g. f (x) for all x in both domains such that g(x) = 0. g(x)
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Similarly, ( f /g)(x) =
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f (x) is continuous at a if f (a) > 0. Note: Since f is continuous at a, the restriction that f (a) > 0 guarantees that, for x close to a, f (x) > 0, and therefore that f (x) is de ned. ONE-SIDED CONTINUITY
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A function f is said to be continuous on the right at a if it satis es conditions (i) (iii) for continuity at a, with lim
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x a+ x a+ x a+
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replaced by lim ; that is, (i) lim f (x) exists; (ii) f (a) is de ned; (iii) lim f (x) = f (a). Similarly, f is continuous on
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CONTINUITY
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[CHAP. 10
the left at a if it satis es the conditions for continuity at a with lim replaced by lim . Note that f is continuous at a if
x a x a
and only if f is continuous both on the right and on the left at a, since lim f (x) exists if and only if both lim f (x) and lim f (x) exist and lim f (x) = lim f (x).
x a x a+ x a x a x a +
EXAMPLES
(a) The function f (x) = x + 1 if x 1 is continuous on the right at 1 (see Fig. 10-3). Note that lim f (x) = lim (x + 1) = x if x < 1 x 1+ x 1+ 2 = f (1). On the other hand, f is not continuous on the left at 1, since lim f (x) = lim x = 1 = f (1). Consequently, f is not continuous at 1, as is evidenced by the jump in its graph at x = 1.
x 1 x 1
(b) The function of example (c) of Section 8.3 [see Fig. 8-1(c)] is continuous on the left, but not on the right at 1. (c) The function of example (b) of Section 9.1 [see Fig. 9-1(b)] is continuous on the right, but not on the left at 1. (d) The function of example (b) of Section 9.2 (see Fig. 9-3) is continuous on the left, but not on the right at 0.
Fig. 10-3 10.3 CONTINUITY OVER A CLOSED INTERVAL
We shall often want to restrict our attention to a closed interval [a, b] of the domain of a function, ignoring the function s behavior at any other points at which it may be de ned. De nition: A function f is continuous over [a, b] if: (i) f is continuous at each point of the open interval (a, b). (ii) f is continuous on the right at a. (iii) f is continuous on the left at b. EXAMPLES
(a) Figure 10-4(a) shows the graph of a function that is continuous over [a, b]. (b) The function f (x) = 2x 1 if 0 x 1 is continuous over [0, 1] [see Fig. 10-4(b)]. otherwise
CHAP. 10]
CONTINUITY
Fig. 10-4
Note that f is not continuous at the points x = 0 and x = 1. Observe also that if we rede ned f so that f (1) = 1, then the new function would not be continuous over [0, 1], since it would not be continuous on the left at x = 1.
Solved Problems
2 x 1 10.1 Find the points at which the function f (x) = x + 1 2 if x = 1 if x = 1 is continuous.
For x = 1, f is continuous, since f is the quotient of two continuous functions with nonzero denominator. Moreover, for x = 1, x2 1 (x 1)(x + 1) = =x 1 x+1 x+1 whence lim f (x) = lim (x 1) = 2 = f ( 1). Thus, f is also continuous at x = 1. f (x) =
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