Contributions to Non-Standard Analysis by W.A.J. and Robinson, A. (Ed.) Luxemburg

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

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Continuous *-meromorphic functions. on on on (52) on on 52, by is d. 52 f d (52) E a E >0 d(z, a ) < 6 f(z) (E E s(f(z), (52, d) E 52 (*S, s), *- 6 >0 8') < E, z or a p f(a). 1. M(s/p). The foIIowing are equivalent: (1) f is S-continuous at zero. ( 2 ) There exist three values a,/3, y E *S,jiniteIy separated in the spherical metric, which f does not attain in the injinitesimals, The magnification o f f , Mf(z), isjinite on the injinitesimals, D. Proof. 11 by on *S on by injinitesimal group by a,/3, y E *S on a ia, a,y E [w, a ; b, c] = [ z , a; /3, y] w-a -w-c - x- z-u z - y , x x 1.

F(Dd(A;E)). 3. Remarks. 5. 8 s 8, h(z) = d(z, a ) > r ) : r > 01, E a E f2 h(z), a ) > ro (z on ro, E *8) S-disk. 4. THEOREM. S-discontinuity or A is an M-set. Proof. 5. Julia-MiIIoux theorem for meromorphic functions. 6. 6. 6. Gavrilov’s classes W,(p 2 a. p 2 1. = by pf(z). W, by IZI’-~ by Ix < - yl < 00. ~lyI’-~. pf(z,,) = a, for a holomorphicfunction deJined in a neighborhood of 00, then for every r > 0 and E > 0 there exists a point such that in the disk < takes on every value in the circle IwI < r with the exception of a set of diameter less than 2/r.

2) (1) (2) (2) (3) d(z,,, y,,) + 0, d(f(x,), f(y,)) w 0 (3) by y, d(x,,, y,,) o E 'N,d(x,, y,) E -= >0 = 0. (3) x,, n d(f(x,,),f(y,,)) > E, by 3. Continuous *-meromorphic functions. on on on (52) on on 52, by is d. 52 f d (52) E a E >0 d(z, a ) < 6 f(z) (E E s(f(z), (52, d) E 52 (*S, s), *- 6 >0 8') < E, z or a p f(a). 1. M(s/p). The foIIowing are equivalent: (1) f is S-continuous at zero. ( 2 ) There exist three values a,/3, y E *S,jiniteIy separated in the spherical metric, which f does not attain in the injinitesimals, The magnification o f f , Mf(z), isjinite on the injinitesimals, D.

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