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2.4 Continuity Objective: Given a graph or equation, examine the continuity of a function, including left-side and right-side continuity. Then use laws of continuity to evaluate a limit. A function is continuous if you can draw it in one motion without picking up your pencil. Conditions of continuity at a point wher e x c : a. f (c) is defined. (a y - value exists at x c) b. lim f ( x) exists. ( the limit exists on right and left) x c c. lim f ( x) f (c) (the y - value is the same as the limit) x c Determine whether the function is continuous at each of the following locations. Explain. 1. 2. 3. 4. 5. x=0 x=1 x=2 x= 4 Over what interval(s) is f continuous? Three types of Discontinuity – removable, jump, non-removable A. Removable Discontinuity (hole) (You can fill the hole by redefining the function at one point) B. Jump Discontinuity (left and right limits are not equal) Three types of Discontinuity – removable, jump, non-removable C. Non-removable “Infinite” Discontinuity (asymptote) 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 -2 -3 -4 -5 – the limit is infinite as c approaches c on one or both sides Left-Continuous and Right Continuous – what does it mean? Section 2.4, Figure 14 Page 69 Example 1 : Let f ( x) x . Compute the left - and right - hand limits at x 0, 1, and 2. Then determine whether f ( x) is left - continuous , right - continuous , or continuous at these points. Example 2 : Compute the left - and right - hand limits at x 0, 1, and 2. Then determine whether f ( x) is left - continuous , right - continuous , or continuous at these points. Example 3 : Find the value of the contant(s) that makes the piecewise function continuous . 2 x 9 x 1 f ( x) 4 x c for x 3 for x 3 Laws of Continuity – page 64 Laws of Continuity – page 64 The Laws of Continuity are intended to help us determine the continuity of functions containing sums, products, quotients, multiples, powers, and roots without looking at a graph. Knowing that a function is continuous allows us to use direct substitution to evaluate a limit. Evaluate each limit using the laws of continuity . 1. lim tan x x 4 2. lim 4 x 2 x 0 3. lim sin 2 ( sin 2 x) x 3 x 1, x 0 4. lim f ( x) if f ( x) 2 x 0 x 1, x 0