Science Mathematics

Analytic Functions by Stanislaw Saks

By Stanislaw Saks

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F) of the point z from the closed set F (cf. § 8, p. 19). F) is therefore continuous on the open plane E0• If the set F reduces to the point oo, then e(z,F)=e(z,oo)=l/izl immediately that the function 38 INTRODUCTION. Theory of sets. and the function e(z,F) is obviously oontinuous on the whole plane E. Hence, there remains to be investigated the case when the set F contains the point oo, but does not reduce to this point. 3) where by min{a,b} we mean, for every pair of real numbers a,b, the smaller of these numbers when the· numbers are different, and their common value when they are equal.

An arbitrary circle whose centre belongs to the set A and which does not contain the point a1 in the interior or on the boundary. Next, let us assume that there have been defined r circles K K 1, 2, • • • , K,. in such a way that the conditions (a), (b), and (c) are satisfied for n�r. Since the circle Kr has a centre at a point belonging to the perfect set A, it contains an infinite number of points of this set. +1 (c) We can therefore determine a circle with centre at the point b, such that conditions (a), (b), and will be satisfied for n=r+l.

Let us assume that the points of the perfect set A {anln=t,2,. IKnln=t,2, , can be arranged in a sequence mine a sequence of circles By induction we deter­ .. • satisfying the following ... conditions for n=l, 2 , ... : (a) KnCKn--1 for n>l, (b) the centre of every circle Kn (c) the closed circle Kn belongs to the set A, does not contain the point K1 With this in view, let us select as an. an arbitrary circle whose centre belongs to the set A and which does not contain the point a1 in the interior or on the boundary.

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