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Continuity and One-Sided Limits – Part 1 Interactive Notes I. What does continuous mean? A function is continuous if there is no ____________________ in the graph (no ___________ or ______________). A function can be continuous at a certain point or it can be continuous on an interval. o Continuity at a Point: A function f is continuous at c if the following three conditions are met. 1. _____________________________________________ Example: The following function is not continuous at x 2 because f (2) is not defined. f ( x) x2 4 x2 2. ______________________________________________ Example: The following function is not continuous at x 3 because lim f ( x ) does not exist. x 3 f ( x) x3 x2 9 3. ______________________________________________ Example: The following function is not continuous at x 4 because although lim f ( x ) exists, x4 lim f ( x) f (4) . x 4 x 2 9, x 4 f ( x) x4 3, o Continuity on an Open Interval: A function is continuous on an open interval if it is continuous at each point in the interval. A function that is continuous on the entire real line (, ) is _________________________________________________________. II. Types of Discontinuities o There are two types of discontinuities: removable and nonremovable. o A discontinuity at c is called _____________________________ if f can be made continuous by appropriately defining or redefining f (c ) . (Removable discontinuities usually appear as _____________.) o Here are some functions that contain removable discontinuities. removable discontinuity at _________ removable discontinuity at _________ removable discontinuity at _________ o A discontinuity at c is called ___________________________________ if f can not be made continuous by appropriately defining or redefining f (c ) . (Nonremovable discontinuities usually appear as ______________________________.) Here are some functions that contain nonremovable discontinuities. nonremovable discontinuity at _________ nonremovable discontinuity at _________ III. How do I detect discontinuities analytically? (Think of _______________________________.) Find the restricted values for the function. If these values can be eliminated by some technique, then the discontinuities are _____________________________; however, if these values cannot be eliminated by some technique, then the discontinuities are __________________________________. o Examples: Discuss the continuity of the following functions. o f ( x) 1 x3 _______________________________________ _______________________________________ o x2 x 6 f ( x) x2 _______________________________________ _______________________________________ o f ( x) x4 x 2 16 _______________________________________ _______________________________________ o f ( x) 2 x 3 _______________________________________ _______________________________________ o 2 x 3, f ( x) 5, x0 x0 _______________________________________ _______________________________________ o 2 x 3, f ( x) x 3, x0 x0 _______________________________________ _______________________________________ o 2 x 3, f ( x) 4, x0 x0 _______________________________________ _______________________________________