chapter5 - Chapter 5 Functions on Metric Spaces and...

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Chapter 5 Functions on Metric Spaces and Continuity When we studied real-valued functions of a real variable in calculus, the techniques and theory built on properties of continuity, differentiability, and integrability. All of these concepts are de ¿ ned using the precise idea of a limit. In this chapter, we want to look at functions on metric spaces. In particular, we want to see how mapping metric spaces to metric spaces relates to properties of subsets of the metric spaces. 5.1 Limits of Functions Recall the de ¿ nitions of limit and continuity of real-valued functions of a real vari- able. De ¿ nition 5.1.1 Suppose that f is a real-valued function of a real variable, p + U , and there is an interval I containing p which, except possibly for p is in the domain of f . Then the limit of f as x approaches p is L if and only if ± 1 ²³ ±² 0 " ± 2 = ± =±²³³±= 0 F ± 1 x ³± 0 ´ ² x ³ p ² ´= " ² f ± x ³ ³ L ² ´ ²³³³ . In this case, we write lim x ´ p f ± x ³ ± L which is read as “the limit of f of x as x approaches p is equal to L.” De ¿ nition 5.1.2 Suppose that f is a real-valued function of a real variable and p + dom ± f ³ .Then fis continuous at p if and only if lim x ´ p f ± x ³ ± f ± p ³ . 183
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184 CHAPTER 5. FUNCTIONS ON METRIC SPACES AND CONTINUITY These are more or less the way limit of a function and continuity of a function at a point were de ¿ ned at the time of your ¿ rst encounter with them. With our new terminology, we can relax some of what goes into the de ¿ nition of limit. Instead of going for an interval (with possibly a point missing), we can specify that the point p be a limit point of the domain of f and then insert that we are only looking at the real numbers that are both in the domain of the function and in the open interval. This leads us to the following variation. De ¿ nition 5.1.3 Suppose that f is a real-valued function of a real variable, dom ± f ² ± A, and p + A ) (i.e., p is a limit point of the domain of f ). Then the limit of f as x approaches p is L if and only if ± 1 ³² ±³ 0 " ± 2 = ± = ±³² 0 ² d ± 1 x ²± x + A F 0 ´ ² x ³ p ² ´= " ² f ± x ² ³ L ² ´³² e ² Example 5.1.4 Use the de ¿ nition to prove that lim x ´ 3 b 2 x 2 µ 4 x µ 1 c ± 31 . Before we offer a proof, we’ll illustrate some “expanded ”scratch work that leads to the information needed in order to offer a proof. We want to show that, corresponding to each ³ 0 we can ¿ nd a = 0 such that 0 ´ ² x ³ 3 ² " n n b 2 x 2 µ 4 x µ 1 c ³ 31 n n ´³ . The easiest way to do this is to come up with a = that is a function of ³ . Note that n n n r 2 x 2 µ 4 x µ 1 s ³ 31 n n n ± n n n 2 x 2 µ 4 x ³ 30 n n n ± 2 ² x ³ 3 ²² x µ 5 ² . The ² x ³ 3 ² is good news because it is ours to make as small as we choose. But if we restrict ² x ³ 3 ² there is a corresponding restriction on ² x µ 5 ² ± to take care of this part we will put a cap on = which will lead to simpler expressions. Suppose that we place a 1 st restriction on = of requiring that = n 1 .I f = n 1 , then 0 ´ ² x ³ 3
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chapter5 - Chapter 5 Functions on Metric Spaces and...

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