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Limits and Continuity 1 Intro to Continuity As we have seen some graphs have holes in them, some have breaks and some have other irregularities. We wish to study each of these oddities. We will use our information of limits to decide if a function is continuous or has holes. 2 Continuity Intuitively, a function is said to be continuous if we can draw a graph of the function with one continuous line. I. e. without removing our pencil from the graph paper. Definition A function f is continuous at a point x = c if 1. f (c) is defined 2. lim f(x) exists 3. lim f(x) f (c) x c x c THIS IS THE DEFINITION OF CONTINUITY 4 Example f (x) = x – 1 at x = 2. a. f (2) = 1 1 x11 b. xlim 2 The limit exist! c. f (2) 1 lim x 1 2 x2 Therefore the function is continuous at x = 2. 5 Example f (x) = (x2 – 9)/(x + 3) at x = -3 a. f (-3) = 0/0 b. Is undefined! x2 9 lim x 3 x 3 -6 -3 The limit exist! c. x2 9 lim f ( 3) x 3 x 3 -6 Therefore the function is not continuous at x = -3. You can use table on your calculator to verify this. 6 Continuity Properties If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0. Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero. 7 Continuity Summary. Functions have three types of discontinuity. Consider 1. Non Removable Discontinuity at vertical asymptote. 2. Removable Discontinuity at hole. 3. Removable discontinuity at jump 8 Worksheet L.4-1 9