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- W2792854054 abstract "{ (x, y): x log I y I = y log I x this latter set consisting of the union of five curves, one of which is the line y=x. The four remaining curves, say C1, C2, C3, and C4 are shown to lie in the first, second, third and fourth quadrants respectively. THEOREM 1 ([1]). In the above situation, (a) any point of C1 is a point of S, (b) the points of S are everywhere dense on C2, C3, and C4 and (c) (-1, 1)CC2, (1, -1)CC4 and these two points are not elements of S. The following questions are then raised. (a) Are there any points of C3 which are not points of S? (b) Are there any points of C2 or C4, apart from those given in Theorem 1 (c), which are not points of S? In this paper we shall prove the following result and thus answer both questions. THEOREM 2. The set S is countably dense on C2, C3, and C4. Further, if (x, y) CS-CC then x and y are both rational or are both irrational. The following question is left unanswered: Does there exist a rational number x and an irrational number y such that (x, y) CS? If so then clearly (x, y) CCl. Proof of Theorem 2. This depends on the following two lemmas the first of which is well known (e.g. [2] p. 47). LEMMA 1. Every uncountable set contains a perfect subset. LEMMA 2. Suppose that f (x) and g(x) are differentiable on some interval I and that f (x) = g(x) on some perfect subset Q of I. Then f'(x) = g'(x) on Q. Proof. Let h(x) =f(x) -g(x). Then if x C Q" @default.
- W2792854054 created "2018-03-29" @default.
- W2792854054 creator A5037412323 @default.
- W2792854054 date "1966-03-01" @default.
- W2792854054 modified "2023-09-24" @default.
- W2792854054 title "The Real Solutions of xν=yx" @default.
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- W2792854054 doi "https://doi.org/10.1080/0025570x.1966.11975694" @default.
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