22 for what points are the functions considered in ch

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22. For what points are the functions considered in Ch. IV , Exs. xxxi dis- continuous, and what is the nature of their discontinuities? [Consider, e.g. , the function y = lim x n (Ex. 5). Here y is only defined when - 1 < x 5 1: it is equal to 0 when - 1 < x < 1 and to 1 when x = 1. The points x = 1 and x = - 1 are points of simple discontinuity.]
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[V : 100] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 216 100. The fundamental property of a continuous function. It may perhaps be thought that the analysis of the idea of a continuous curve given in § 98 is not the simplest or most natural possible. Another method of analysing our idea of continuity is the following. Let A and B be two points on the graph of φ ( x ) whose coordinates are x 0 , φ ( x 0 ) and x 1 , φ ( x 1 ) respectively. Draw any straight line λ which passes between A and B . Then common sense certainly declares that if the graph of φ ( x ) is continuous it must cut λ . If we consider this property as an intrinsic geometrical property of con- tinuous curves it is clear that there is no real loss of generality in supposing λ to be parallel to the axis of x . In this case the ordinates of A and B cannot be equal: let us suppose, for definiteness, that φ ( x 1 ) > φ ( x 0 ). And let λ be the line y = η , where φ ( x 0 ) < η < φ ( x 1 ). Then to say that the graph of φ ( x ) must cut λ is the same thing as to say that there is a value of x between x 0 and x 1 for which φ ( x ) = η . We conclude then that a continuous function φ ( x ) must possess the following property: if φ ( x 0 ) = y 0 , φ ( x 1 ) = y 1 , and y 0 < η < y 1 , then there is a value of x between x 0 and x 1 for which φ ( x ) = η . In other words as x varies from x 0 to x 1 , y must assume at least once every value between y 0 and y 1 . We shall now prove that if φ ( x ) is a continuous function of x in the sense defined in § 98 then it does in fact possess this property. There is a certain range of values of x , to the right of x 0 , for which φ ( x ) < η . For φ ( x 0 ) < η , and so φ ( x ) is certainly less than η if φ ( x ) - φ ( x 0 ) is numerically less than η - φ ( x 0 ). But since φ ( x ) is continuous for x = x 0 , this condition is certainly satisfied if x is near enough to x 0 . Similarly there is a certain range of values, to the left of x 1 , for which φ ( x ) > η . Let us divide the values of x between x 0 and x 1 into two classes L , R as follows: (1) in the class L we put all values ξ of x such that φ ( x ) < η when x = ξ and for all values of x between x 0 and ξ ; (2) in the class R we put all the other values of x , i.e. all numbers ξ
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[V : 100] LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE 217 such that either φ ( ξ ) = η or there is a value of x between x 0 and ξ for which φ ( x ) = η . Then it is evident that these two classes satisfy all the conditions im- posed upon the classes L , R of § 17 , and so constitute a section of the real numbers. Let ξ 0 be the number corresponding to the section.
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