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Unformatted text preview: MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.5: Continuity and continuous functions So, youd like to be able to work only with all of those nice functions for which the Method of Direct Substitution works, huh? Good news! Those are the functions well be dealing with most of the time. Theyre the ones without jumps or gaps in the places where theyre defined, and often they dont even have asymptotes, at least at the points well be looking at. They are precisely the functions. Definition. A function f is called at a number a if lim x a f ( x ) = . Notice there are really three things going on here: 1. the limit lim x a f ( x ) exists, 2. the function f is defined at the point x = a , and 3. the two values lim x a f ( x ) and f ( a ) are . If f is not continuous at a point x = a , we say that f has a at that point. There are three types of that may arise, and weve seen examples of them all: 1. discontinuities are those at which the function would be con- tinuous, if only we could redefine it to take a certain value at the given point. That is, the limit exists at the point x = a , and the hole in the graph of f could be removed by redefining f ( a ) = . An example of such a function is f ( x ) = x 2 + x- 6 x- 2 = ( x +3)( x- 2) x- 2 at the point x = 2. You should sketch this functions graph in the space below and see why it is of this sort: 2.2....
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- Fall '07