Stewart6e2_5 - MATH 191 Sections 1 and 3 Calculus I Fall...

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MATH 191, Sections 1 and 3 Calculus I Fall 2007 Section 2.5: Continuity and continuous functions So, you’d 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 we’ll be dealing with most of the time. They’re the ones without jumps or gaps in the places where they’re defined, and often they don’t even have asymptotes, at least at the points we’ll 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 we’ve 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
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