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

ISBN-10: 0720420652

ISBN-13: 9780720420654

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On is called approximately continuous at a point xo E I whenever for every E > 0 the point xo is a point of density one of the set {x:If ( x ) - f(xo)l < E } . by 1 on R on e, f almost everywhere approximately continuous on e. f bounded, then f is approximately continuous at a point x E e if and only if the function F(x) = f(t)dt is direrentiable at x and its derivative F ( x ) at x satisfies F ( x ) = f (x). e e R, e is approximately continuous at x i f and only if x is a point of density one of e.

S. 11. 3. , 0 W t h e conditionsof the theorem are equivalent to (standard) uniform continuity of the f k on all of 8. 8 [1965]. 4. Remark. 5. on b E B w E B w(a) = b. W B. 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)]).

F A,, B, E (A, B) A4, B4 E ( A 3 , B , ) M d ( f ( x ) ,f ( y ) ) w 0. 0 by (An)nEN w C = C C A, = B,. (A by 6, ( E d(f(A,), f ( C ) ) M A(f(A,), f(B,)), 6 E d(A,, C ) + d(C, B,) of (1) = d(A,, B,). (BJnsN. MEROMORPHIC FUNCTIONS Remarks. f ( x ) = x on 51 by 11 s(f(O), M(s/x)f(x) = 1 f(l))/x 1) = +4(2/J2) = 47cJ2 > 1. 3. 1. 2. 3. 1 IIx - yII. 4. 2. 4. COROLLARY. I f f : ( X , d)+ (Y, A ) is an internal map whose magn$cation exists and is finite at each point of the internal complete convex metric space ( X , d), then is uniformily S-continuous on X .

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

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