Lectures on the Theory of Functions of Real Variables Volume N . 2
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Lectures on the Theory of Functions of Real Variables Volume N . 2

Lectures on the Theory of Functions of Real Variables Volume N . 2


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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. 1912 Excerpt: ...of lengtli lt in the middle of 21. In a similar manner let us place in each of the remaining or white intervals, a black interval, whose total lengths = ia. Let us continue in this way; we get an enumerable set of black intervals g, and obviously % =. If we omit the end points from each of the black intervals we get a set SB, and obviously S = X. The set $ = 21-S3 we call a Harnack set. This is complete by 324; and by 338, 347, $ = $ = I--. When = I, !q is discrete, and the set reduces to a set similar to Cantor's set. When A. I, we get an apantactic perfect set whose upper content is I--0, and whose lower content is 0. 2. Within each of the black intervals let us put a set of points having the end points for its first derivative. The totality of these points form an isolated set $ and = But by 331, $ = If now $ is not discrete, 3? is not. We have thus, the theorem: There exist isolated point sets which are not discrete. 3. It is easy to extend Harnack sets to 9? . For example, in 9?2, let S be the unit square. On two of its adjacent sides let us place congruent Harnack sets . We now draw lines through the end points of the black intervals parallel to the sides. There results an enumerable set of black squares (c) = 3n. The sides of the squares (c) and their limiting points form obviously an apantactic perfect set Let a + a+... = m be a series whose sum 0 m 1. We can choose $ such that the square corresponding to its largest black interval has the area a; the four squares corresponding to the next two largest black intervals have the total area a, etc. Then -2o- = m. Hence 1=1-m = S. 355. 1. If = em is an enclosure of 21 such that 2.-f. it is called an e-enclosure. Let A be the complement of 21 with respect to the m


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Product Details
  • ISBN-13: 9781152996519
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 96
  • Spine Width: 5 mm
  • Width: 189 mm
  • ISBN-10: 1152996517
  • Publisher Date: 01 May 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 186 gr


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