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Unformatted text preview: 174 CHAPTER 11 Continuity Consider the problem of measuring the side length of a square and then using the measured date to compute the area of the square. For example, if one side is measured to be 2 . 74 inches, then the area would be computed as (2 . 74) 2 = 7 . 5076 square inches. We know, of course, that no measurement is precise. There will undoubtedly be small errors in the measurement of the side length. This will result in errors in the computed area. We also know from experience that the errors in the area computation will be small as long as the errors in the side measurement are also small; values of s close to 2 . 74 will produce values of A close to (2 . 74) 2 . In mathematical terms, this amounts to saying that lim s → 2 . 74 s 2 = (2 . 74) 2 . This is a statement of the fact that the area function A ( s ) = s 2 is continuous at the point s = 2 . 74. In general, we define continuity as follows: Definition 1 . A function f is continuous at a provided that lim x → a f ( x ) = f ( a ) . Implicit in the above definition is the requirement that a belong to the domain of f . It is also implicit that f ( x ) be defined for all x sufficiently close to a since otherwise, the limit would not exist. (We must be able to get  f ( x ) − f ( a )  < ǫ for all x between a − δ and a + δ .) Thus, the above definition requires that there is a δ > 0 such that the interval ( a − δ, a + δ ) belongs to the domain of f . This, however, presents us with a difficulty. According to this definition, the function y = √ x is not continuous at x = 0: lim x → √ x 175 176 11. CONTINUITY does not exist since √ x is not defined for x < 0. The best we can say is that lim x → + √ x = 0 . Because of such examples, we are forced to amend our definition in the case that the domain of f ( x ) is a closed (or halfclosed) interval. Before doing so, however, we first give the “official” definition of left and right hand limits: Definition 2 . Let f ( x ) be a function. We say that lim x → a + f ( x ) = L provided that for all ǫ > 0 there is a δ > 0 such that  f ( x ) − L  < ǫ for all x satisfying <  x − a  < δ, x > a. The definition of lim x → a f ( x ) = L is identical, except that “ x > a ” in the last inequality above is replaced by “ x < a .” Now our amended definition of continuity states: Definition 3 . Suppose that the domain of f ( x ) is the interval [ a, b ]. We say that f ( x ) is continuous at a if lim x → a + f ( x ) = f ( a ) We say that f ( x ) is continuous at b if lim x → b f ( x ) = f ( b ) . Intuitively, continuity may be described in the same terms as we did for the square function: values of x near a produce values of f ( x ) near f ( a ). (See Figure 1). It is very fortunate that most of the functions which arise in the real world are continuous. Otherwise, we would never be able to calculate anything!...
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 Spring '08
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 Calculus, Topology, Continuity, Intermediate Value Theorem, Continuous function

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