1.8 - 1.8 continuity Dr Chenying Wang Lecturer Department...

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Unformatted text preview: 1.8 continuity Dr. Chenying Wang Lecturer Department of Mathematics Outline Definition of continuity How to determine where a function is continuous Intermediate Value Theorem 2 Only for math 140, sections 1 & 020, fall 15 Definition of Continuity A function is continuous at a number a if lim f(x) = f (a): x!a This de¯nition implicitly requires that a) f (a) is de¯ned b) limx!a f (x) exists c) limx!a f (x) = f (a). We say that f is discontinuous at a (or f has a discontinuity at a) if f is not continuous at a. Note: f (x) ia continuous at a means that inputs near a are sent to outputs near f(a). The graph of a continuous function is a single curve which can be drawn without lifting the pen from the paper. 3 Only for math 140, sections 1 & 020, fall 15 Example 1 The following figure shows the graph of a function f. At which numbers is f discontinuous? Why? 4 Only for math 140, sections 1 & 020, fall 15 Example 1 - Solution Solution: f(x) is discontinuous at ¡3 because f (¡3) is not de¯ned. f(x) is discontinuous at 0 because limx!0 f (x) does not exist. f(x) is discontinuous at 2 because limx!2 f (x) 6= f (2) f(x) is discontinuous at 6 because limx!6 f (x) does not exist. ( f (x) has an in¯nite lmit at 6.) Note: -3 and 2 are called removable discontinuities. 0 is called a jump discontinuity. 6 is called an infinite discontinuity. 5 Only for math 140, sections 1 & 020, fall 15 Three types of discontinuities There are three types of discontinuities. (a) A discontinuity is removable if we could remove the discontinuity by redefining the function at a single number. (b) A discontinuity is an infinite discontinuity if it occurs at a vertical asymptote. (c) A discontinuity is a jump discontinuity if it occurs as a function "jumps" from one value to another. 6 Only for math 140, sections 1 & 020, fall 15 Example 2 Where are each of the following functions discontinuous? x2 ¡ x ¡ 2 (a): f(x) = x¡2 1 (b): f(x) = x 8 < x2 ¡ x ¡ 2 if x 6= 2 (c): f(x) = : 1 x¡2 if x = 2 (d): f(x) = x (the greatest integer function) ½ sin(¼x) if x < 1 2 (e): f(x) = 2x if x ¸ 1 2 7 Only for math 140, sections 1 & 020, fall 15 Example 2(a) - Solution Solution: Since f(2) is not defined, f(x) is discontinuous at 2. Since x2 ¡ x ¡ 2 lim f (x) = lim x!2 x!2 x¡2 (x ¡ 2)(x + 1) = lim x!2 x¡2 = lim x + 1 x!2 =3 exists, 2 is a removable discontinuity. 8 Only for math 140, sections 1 & 020, fall 15 Example 2(b) - Solution Solution: 1 lim f (x) = lim x!0 x!0 x DNE and f(0) is not de¯ned. So f is discontinuous at 0 and 0 is an in¯nite discontinuity. 1 (b): f(x) = x 9 Only for math 140, sections 1 & 020, fall 15 Example 2(c) – Solution Solution: f(2) = 1 is de¯ned and x2 ¡ x ¡ 2 lim f (x) = lim x!2 x!2 x¡2 (x ¡ 2)(x + 1) = lim x!2 x¡2 = lim x + 1 x!2 =3 exists. But limx!2 f(x) 6= f (2), so f is not continuous at 2 and 2 is a removable discontinuity. 10 Only for math 140, sections 1 & 020, fall 15 Example 2(d) - Solution Solution: The greast integer function f(x) = x has discontinuities at all of the integers because limx!n x does not exist if n is an integer. All the integers are jump discontinuities. 11 Only for math 140, sections 1 & 020, fall 15 Example 2(e) - Solution Solution: Since lim¡ f(x) = lim+ f(x) = 1; x! 1 2 x! 1 2 we have limx! 1 f(x) = 1. 2 1 Since f( 2 ) = 1, we have limx! 1 f (x) = f( 1 ). So f 2 2 does not have any discontinuities. ½ (e): f(x) = 12 sin(¼x) if x < 2x if x ¸ 1 2 1 2 Only for math 140, sections 1 & 020, fall 15 Definition A function f is continuous from the right at a if lim+ f (x) = f (a) x!a and f is continuous from the left at a if lim f (x) = f (a) ¡ x!a Note: In example 2(d), the function f is continuous from the right at all the integers. 13 Only for math 140, sections 1 & 020, fall 15 Definition A function f is continuous on an interval if it is continuous at every number in the interval. Note: If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left. In example 2(d), the function f is continuous on [n, n+1) for any integer n. 14 Only for math 140, sections 1 & 020, fall 15 Example 3 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. 8 if x · 0 < 1 ¡ x2 2¡x if 0 < x · 2 : f(x) = : (x ¡ 2)2 if x > 2 15 Only for math 140, sections 1 & 020, fall 15 Theorem If f and g are continuous at a and c is a constant, then the f functions f + g, f ¡ g, cf, f g, and (if g(a) 6= 0) are also g continuous at a. The following types of functions are continuous at every number in their domains: polynomials rational functions root functions trigonometric functions 16 Only for math 140, sections 1 & 020, fall 15 Example 4 Use continuity to evaluate the limit. p ( 16x + 3) tan(¼x) lim x2 x! 1 4 17 Only for math 140, sections 1 & 020, fall 15 Example 5 On what intervals is each function continuous? (a): (b): (c): (d): (e): (f ): (f ): (g): 18 2015 9 1 3 f (x) = 99x ¡ 3x ¡ x + 7 x5 + 3x4 ¡ x2 + 100 f (x) = 4x2 ¡ 9 p f (x) = 4x2 ¡ 9 1 f (x) = p 4x2 ¡ 9 p p x3 + cos2 (2x) sin x f (x) = 3 x + 1 ¡ + 2 + 2 x2 ¡ x ¡ 6 3x + 2 f (x) = 8 cot x 1 p f (x) = 16 ¡ x ¡ 1 f (x) = cos(sin(x3 ¡ 10)) Only for math 140, sections 1 & 020, fall 15 Theorem If f is continuous at b and limx!a g(x) limx!a f(g(x)) = f(b). In other words, = b, then lim f(g(x)) = f(lim g(x)): x!a x!a If g is continuous at a and f is continuous at g(a), then the composite function (f ± g)(x) = f(g(x)) is continuous at a. Note: The above theorem is often expressed informally by saying “ a continuous function of a continuous function is a continuous function. 19 Only for math 140, sections 1 & 020, fall 15 Example 6 For what value of the constant c is the function f continuous on (¡1; 1)? ½ 2 x + cx if x < 3 f(x) = cx2 ¡ 3 if x ¸ 3 20 Only for math 140, sections 1 & 020, fall 15 Intermediate Value Theorem If f (x) is continuous on the closed interval [a; b] and let N be any number between f(a) and f(b), where f(a) 6= f (b). Then there exists a number c in (a; b) such that f (c) = N. Note: The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values f (a) and f (b). 21 Only for math 140, sections 1 & 020, fall 15 Example 7 Use the Intermediate Value Theorem to show that there is a root of the equation p 3 x =1¡x in the interval (0; 1). 22 Only for math 140, sections 1 & 020, fall 15 ...
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