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1003 Continuity General Idea: ______________________________________________________________________________ We already know the continuity of many functions: Polynomial (Power), Rational, Radical, Exponential, Trigonometric, and Logarithmic functions DEFN: A function is continuous on an interval if it is continuous at each point in the interval. DEFN: A function is continuous at a point c IFF a) __________________________________________ _____________________________________ b) __________________________________________ _____________________________________ c) __________________________________________ _____________________________________ THEOREM: Note: All three steps are implied. One sided Continuity: A function y = f(x) is continuous at a , the left endpoint a of its domain if lim f ( x) f (a ) . A function y = f(x) is continuous at b , the right endpoint a of its domain if lim f ( x) f (b) . x a xb Theorem: A function is Continuous on a closed interval if it is continuous at every point in the open interval and continuous from one side at the end points. Example : The graph over the closed interval [-2,4] is given. Discontinuity Continuity may be disrupted by: (a). c c (b). c (c). c Removable or Essential ( Non-Removable) Discontinuities __________________________________________________________________________________________ __________________________________________________________________________________________ Examples: Identify the x-values (if any) at which f(x) is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? f ( x) x 2 x4 f ( x) 1 ( x 3) 2 3 x, x 1 f ( x) 3 x, x 1 Identify the x-values (if any) at which f(x) is not continuous. Identify the reason for the discontinuity and the type of discontinuity. Is the discontinuity removable or essential? Algebraic Method 3x 2 x 2 f ( x) 2 3x 4 x 2 Reason Type x = -3 ____________________ ________________ x = -2 ____________________ ________________ x=0 ____________________ ________________ x=1 ____________________ ________________ x=2 ____________________ ________________ x=3 ____________________ ________________ 1- x 2 x 1 2 f ( x) x - 2 1 x 3 x2 9 x3 x 3 Find c that makes f ( x) continuous. 4 x 2c x 5 f ( x) 2 x c x 5 Consequences of Continuity: A. INTERMEDIATE VALUE THEOREM ** Existence Theorem _______________________________________________ _______________________________________________ _______________________________________________ c _______________________________________________ EX: Verify the I.V.T. for f (c). Then find c. f ( x) x 2 on f (c) 3 [1, 2] EX. Zero Locater Corollary. Show that the function f ( x) x3 2 x 1 has a ZERO on the interval [0,1]. ( CALCULUS AND THE CALCULATOR: The calculator looks for a SIGN CHANGE between Left Bound and Right Bound.) EX. Sign on an Interval - Corollary (Number Line Analysis) 1 3 x 1 2 ( x 1)( x 2)( x 4) 0 B. EXTREME VALUE THEOREM On every closed interval there exists an absolute maximum value and minimum value y x . Assignment 1003: FDWK p. 84 11) a) does f(-1) exist? b) does lim f ( x) exist? x1 c) does lim f ( x) f ( 1) ? x1 d) Is f(x) continuous a t f(-1)? 12) a) does f(1) exist? b) does lim f ( x) exist? x1 c) does lim f ( x) f (1) ? x1 d) Is f(x) continuous a t f(1)? 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it possible to extend f to make it continuous at x = 0 ? Explain. 18) Is it possible to extend f to make it continuous at x = 3 ? Explain. (a) Find each point of discontinuity. Give the reason for your answer. (b) Which of the discontinuities are removable? essential? 19) 3 x , x 2 f ( x) x 2 1 , x 2 20) 47) Find a value for a so that the function is continuous. x2 1 , x 3 f ( x) 2ax , x 3 49) Find a value for a so that the function is continuous. 4 x 2 , x 1 f ( x) 2 ax -1 , x 1 3 x , x 2 f ( x) 2 , x2 x , x2 2 21) 1 , x 1 f ( x) x 1 x3 2 x 5 , x 1 48) Find a value for a so that the function. is continuous 2x 3 , x 2 f ( x) ax 1 , x 2 50) Find a value for a so that the function. is continuous x2 x a , x 1 f ( x) 3 , x 1 x