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solutions_09_18 - Math 1a Continuity(Solutions 1 At which...

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Math 1a: Continuity (Solutions) September 18, 2009 1. At which values of x are the following functions continuous? 1. a ( x ) = b x c , the greatest integer less than or equal to x . For all x except the integers: . . . , - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 , . . . . 2. b ( x ) = the taxi fare (in dollars) for distance x (in miles). Assume that the meter ticks every 0 . 1 miles. Note that the function is defined only for x > 0. It is continuous for all x > 0 except x = 0 . 1 , 0 . 2 , 0 . 3 , . . . . 3. c ( x ) = sin( x ) + cos( x ) . Continuous for all x . 4. d ( x ) = | x | , the absolute value of x . Continuous for all x . 2. A phone plan states that “Charges are calculated on a per minute basis, and rounded up to the next whole minute”. Is the price of a phone call a continuous function of its length? Solution. No, it is not. The price function is discontinuous at time t = 1 min , 2 min , 3 min , . . . . 3. Define the function f ( x ) by f ( x ) = xe cos(1 /x ) for x 6 = 0 . How should I define f (0) to make f continuous? Solution. For the function to be continuous, f (0) should be defined so that f (0) = lim x 0 f ( x ) .
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