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1.4 Continuity Calculus Section 1 Understanding Continuity 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. Definition of Continuity A function is continuous at a point c if the following three conditions are met. 1. 2. 3. f(c) is defined lim f ( x ) exists xc lim f ( x ) f ( c ) x c Definition of Continuity A function is continuous on an open interval (a, b) if it is continuous on each point in the interval. A function that is continuous on the entire real line is everywhere continuous. f(x) is continuous on (-3,2) One Sided Limits and Continuity A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a, b) and lim f ( x ) f ( a ) and lim f ( x ) f (b ) xa x b The function is continuous from the right at a and continuous from the left at b. f(x) is continuous on [-3,2] Example 1: Continuity of a Function Discuss the continuity of each function. a. b. c. d. x2 1 f ( x) x 1 1 g ( x) x x 1, x 2 h( x ) 2 x 1, x 2 x 1, x 0 i( x) 2 x 1, x 0 7 Example 1a Solution There is a removable discontinuity at x = 1 because f(1) is not defined (Definition of Continuity Condition 1). x2 1 f ( x) x 1 8 Example 1b Solution g ( x) 1 x The function has a nonremovable discontinuity at x = 0 because lim g ( x) x 0 does not exist (Definition of Continuity Condition 2). 9 Example 1c Solution The limit from the right of x = 2 does not equal the limit from the left. Therefore, the limit as x approaches 2 does not exist. Function has a discontinuity at x = 2 because lim g ( x) does not exist x 0 (Definition of Continuity Condition 2). x 1, x 2 h( x ) 2 x 1, x 2 10 Example 1d Solution The function is continuous on the entire real line. x 1, x 0 i ( x) 2 x 1, x 0 11 Try These Discuss the continuity of each function. x2 4 a. f ( x) x2 1 b. g ( x) x3 x 2, x 5 c. h( x) 3 x 8, x 5 x 4, x 0 d . i ( x) 2 x 1, x 0 12 Removable Discontinuities: (You can fill the hole.) Nonremovable Discontinuities: jump infinite oscillating “Discussing Continuity” Continuous or discontinuous? If discontinuous Removable or nonremovable discontinuity? At what x-value is the discontinuity? What condition is not met? (See slide 4 for conditions) Condition 1: f(c) is defined - means the function exists when x =c Condition 2: limit as x approaches c of f(x) existis – means that the “two sides of the functions meet at c,” no “jumps, or asymptotes Condition 3: Limit and value are equal – means the is no “hole” with a “dot” filled in elsewhere 14 Continuity by Function Type Polynomials are everywhere continuous Sine and Cosine are everywhere continuous Rational functions and other trig functions are continuous except at xvalues where their denominators equal zero. “Removable” discontinuity if factoring and canceling “removes” the zero in the denominator “Non-removable” otherwise. (Recall that vertical asymptotes occur where numerator is nonzero and the denominator is zero.) Root functions are continuous, except at x-values that would result in a negative value under an even root For piecewise functions, find the f(x) values at the x-value separating the regions of the function. If the f(x) values are equal, the function is continuous. Otherwise, there is a (non-removable) discontinuity at this point. 15 Properties of Continuity 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 = 3x + cos(x) Section 2 Finding the Intervals on Which a Function is Continuous 17 Examle 2: Describe the interval(s) on which f is continuous. f ( x) x x 3 Solution: The function is continuous on the interval [-3, ) 18 Try This: Find the interval on which f is continuous. x2 2 x, f ( x) 2 x 4 x 1, x 2 Solution: f(x) has a non – removable discontinuity at x = 2. The intervals on which f is continuous are (- , 2] and [2, ) 19 Section 3 The Intermediate Value Theorem 20 Suppose this pool holds 22,000 gal. of water. Then there is some point in time, when it was being filled that it held exactly 15,000 gal. It could not have skipped over that number. 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 Here is a formal statement of the Theorem: 23 Example 3: Intermediate Value Theorem Without graphing, show that f(x) = 2x4 -3x2 + 5x +2 has at least one zero between -2 and -1. Solution: This function is continuous on the interval [-2, -1] (and everywhere else for that matter), and f(-2) =12 f(-1) = -4 Therefore somewhere between x = -2, and x = -1, f(x) passes through 0. 24 Try This For the function f(x) = x2 -6x + 8 , a) use the Intermediate Value Theorem to show that the function has a zero on the interval [0, 3] b) find the value of "c" guaranteed by the theorem such that f(c) = 0. 25 Solution Verify that the Intermediate Value theorem applies on [0, 3] and find the value of "c" guaranteed by the theorem. f(c) = 0 f(x) = x2 -6x + 8 f(0) = 8 f(3) = -1 0 is between -1 and 8. f(c) = 0 c2 – 6c + 8 = 0 c = 2 or 4 Not 4, why? 26