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By W.A.J. and Robinson, A. (Ed.) Luxemburg

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Extra resources for Contributions to Non-Standard Analysis

Example text

N - Let f:U -+ S be a continuousfunction. 1. continuous on U ifand only iffor everypair of equivalent subsets U, A , A,, the cluster sets C ( f ; A , ) and C ( f ; A , ) are equal. ProoJ: (+): A,) = C ( f ; A,). n *A,)) = n * A j ) )= is (Yn)nsN q(xn, Y n ) a # p cf(yn). 2. + 0 - ~(0, xn) { x n : nE N } -+ OD, {y,}, ~(0, Vn) C(f; -+ 2 a -, # CCf; {A,}).

A. J. LUXEMBURG 32 {f} p 2 2. 20 {fa} E C(T) 1 Ip < 2. xE 1

22 E *T) = 0, {E(n) : n E * Z } - *f, 2 IF(n) - n@Z A IF(t) = 2n - *f (t)I2 = 0. 13 cnnZ - A = 0. W. A. J. 4. (Hausdorfs-Young Inequalities). 10. 1 1 (s,,,)~. m. IIsmllq by p. 6. f

6. Julia exceptional functions. 3 w(1) = z. x lzl, n(z- 2")/(z + 2"), < TC f a = 1. 7. Yosida's Theory [1934]. 8 = = {z + b:b E B}, d(x, y ) = Ix - yI a = 0. 1 M(s/d) (z)= P ! 8. Normal meromorphicfunctions. ) B = U = {z E C:IzI < l}, [(z - ct)/(Ez - 1)]:O E R, lcl[ < 11, x = U,a = 0 d(x, y ) = ~ ( xy ,) , IJx - lj)/(lx - yI - IJz - ll)]). lf(Z> = w(0) = z. )w(O)lPf(z) = (1 -zmf(z), K. D. STROYAN 58 x x 0. (X, (U,q). *U, q-galaxy G(CI) = { z E * U : q ( z ,CI) CI E CI, 01 E 01. 9. Ifcp(t), 0 5 t < 1, is a continuous curve in U with Iq(t)l + 1, and normal function f satisfies f(cp(t)) -, b as t --t 1, then f o W has the constant function b as a limit.

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