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Continuity 2.3 Quick Algebra Review How can you tell whether a graph has a vertical asymptote or a hole? x2 f ( x) 2 x 3x 10 f ( x) x 2 ( x 5)( x 2) There is a vertical aysmptote when x 5. There is a hole when x 2. (2, ) Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. Continuous Not continuous A function is continuous at a point if the limit is the same as the value of the function. lim f ( x) must exist and be equal to f (c) xc Is the function continuous at each value of x? Explain your answer. x 1 x2 x3 2 1 1 2 3 4 x0 x4 The function has one-sided continuity at x=0. Is the function continuous at x = -3? x 19 4 , x 3 f ( x) x 3 1 , x 3 8 lim f ( x) must equal f ( 3) x 3 Types of Discontinuity (removable discontinuity) Jump Discontinuity is typically caused by a piece-wise function whose pieces don’t meet. The limit does not exist because the left-hand limit does not equal the right-hand limit. Example: x 3, f ( x) x 1, x0 x0 Find the real number, c that makes the function everywhere continuous. x 2, f ( x) 2 x c, x3 x3 y x2 8 (3,1) y x2 Point Discontinuity occurs when a function has a hole. The limit exists but it does not equal the function value. Example: x 2 11x 28 f ( x) x4 x 2 11x 28 lim ( x 4)( x 7) lim( x 7) 3 lim x4 x 4 x 4 x4 x4 There is a hole at (-4,3). Point Discontinuity is also called Removable Discontinuity. To remove the discontinuity, just “fill in the hole”. Make the last problem everywhere continuous. x 2 11x 28 f ( x) x4 There is a hole at (-4,3). x 2 11x 28 , x 4 f ( x) x4 x 4 3, Infinite/Essential Discontinuity occurs when there is a vertical asymptote. The limit does not exist because it is either or . Example: 3x f ( x) x2 Example: Give all x-values for which the function is discontinuous, and classify each instance of discontinuity. 9 x 2 3x 2 (3x 1)(3 x 2) f ( x) 2 3x 13x 4 (3x 1)( x 4) There is an infinite discontinuity (vertical asymptote) at: There is a removable discontinuity (hole) at: ( ) Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. Also: Composites of continuous functions are continuous. examples: y sin x 2 y cos x Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between f a and f b . f b Because the function is continuous, it must take on every y value between f a and f b . f a a b Use the Intermediate Value Theorem to explain 2 why the function f ( x) x 3x 6 must have a root (x-intercept) on the closed interval [1,2]. f (1) f (2) If f (1) 2 and f (2) 4, then by the Intermediate Value Thm, the function must take on all y values between -2 and 4, so a y value of 0 must exist and it will happen when x is between 1 and 2. You can use your calculator to find a better approximation of the root. (1.372) Example 5: Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists.) f 1 1 x x3 1 0 x3 x 1 Guess and check. f 2 5 Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. To find a better approximation, use your calculator to find the zeros. f x x3 x 1 1.32472 pg 80 #11-28,35-41 (no calc) Wkst: Continuity