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The Theory of Functions of a Real Variable and the Theory of Fourier's Series: (English)

The Theory of Functions of a Real Variable and the Theory of Fourier's Series: (English)

          
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About the Book

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1907 Excerpt: ...rectangles enclosing all the points of the arc have in common has a lower limit greater than zero. The two continuous functions f(t), 4 (t), which define x, y as functions of ty possess for no value of t definite differential coefficients, and are perhaps the simplest examples of continuous non-differentiable functions. 248. It might at first sight appear that a curve having the same properties as that of Peano might have been defined by restricting t--'axa... to be such that an infinite number of digits other than 0 are present, and then defining x, y by x = 'axaab..., y='a2a4a6.... If however the double representation of x, y were not restricted, as in the case of t, there would be no value of t corresponding to, say, #=1000..., 2/ =-2000.... If (x, y) were on the other hand so restricted, there would be no values of (xy y) corresponding, for example, to $ = 111010101.... It thus appears that some such rule as that given by Peano is necessary to obviate the difficulty caused by the double representation of a certain class of rational numbers, in a given scale. The method may easily be extended to obtain a continuous correspondence between the points in a cube and those in a linear interval. A somewhat different method of establishing correspondence between the points of the square, and those of the linear interval, is the following: ---Let tx denote one of the perfect set of points defined by Qj Qj2 Ct3 when every a is either 0 or 2. For such a point tu oo and y may be denned by I/O;.o., a., 2V2 22 + 23+ A point t which does not belong to the perfect set is interior to one of the See Lebesgue, Legons sur Vintegration, p. 44. complementary intervals ( /, /') of the set; in such an interval we may define #, 2/ as linear functions of t, thus where ...


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Product Details
  • ISBN-13: 9781236294487
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 288
  • Series Title: English
  • Weight: 517 gr
  • ISBN-10: 1236294483
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Spine Width: 15 mm
  • Width: 189 mm


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