D:\User\sbornik_38_pdf\29.DVI Mathematical Problems of Computer Science 38, 72, 2012. E xtensions of M ar kov's Constr uctive Continuum and Unifor m Continuity of Constr uctive Functions B o r is A . K u s h n e r Department of Mathematics University of Pittsburgh at Johnstown Johnstown W e c o n s id e r e ve r ywh e r e d e ¯ n e d c o n s t r u c t ive fu n c t io n s ( c .f.) o n t h e c lo s e d u n it c o n s t r u c - t ive in t e r va l. A s is we ll-kn o wn b y t h e fa m o u s Za s la vs ky-Ts e it in Th e o r e m s u c h a c .f. c a n b e e ®e c t ive ly n o n u n ifo r m ly c o n t in u o u s . In t h is c a s e it c a n n o t b e e xt e n d e d t o a c la s s ic a l c o n t in - u o u s fu n c t io n . In r e a lit y, in e ve r y kn o wn c o u n t e r -e xa m p le a s in g u la r it y c o u ld b e d is c o ve r e d a lr e a d y o n t h e le ve l o f p s e u d o n u m b e r s . L e t u s r e c a ll t h a t a p s e u d o n u m b e r is a r e c u r s ive s e - qu e n c e o f r a t io n a ls t h a t is a Ca u c h y s e qu e n c e c la s s ic a lly. P s e u d o n u m b e r s c a n b e c o n s id e r e d a s Á0 ( ¢ 2 ) -c o m p u t a b le n u m b e r s a s we ll. L e t D b e t h e s e t o f a ll c o n s t r u c t ive r e a l n u m b e r s ( Ma r ko v's Co n t in u u m in t h e t it le ) , D1 t h e s e t o f a ll p s e u d o n u m b e r s . A c .f. f is s a id t o b e 1 -c o m p le t e if it c a n b e e xt e n d e d t o a c o m p u t a b le ( a n d s o c o n t in u o u s ) fu n c t io n o ve r D1. T heor em 1 There is a 1-complete c.f. that is e®ectively nonuniformly continuous. Th is r e s u lt is r a t h e r p r e c is e a s a c .f. c o n t in u o u s ly e xt e n d ib le t o Á00-c o m p u t a b le n u m b e r s is u n ifo r m ly c o n t in u o u s c la s s ic a lly. A s is we ll kn o wn e ve r y c .f. c a n b e c o m p u t e d o n D b y a K le e n e o p e r a t o r ( p a r t ia l-r e c u r s ive o p e r a t o r ) . Th e fo llo win g r e s u lt t o g e t h e r wit h Th e o r e m 1 s h o ws t h a t t h e r e is a n e s s e n t ia l d i®e r e n c e b e t we e n Ma r ko v's a n d K le e n e 's Co m p u t a b ilit y o ve r D1. T heor em 2 A c.f. f is constructively uniformly continuous i® there is a 1-complete K leene operator that computes f. R e fe r e n c e s [1 ] B .A . K u s h n e r , S o m e E xt e n s io n s o f Ma r ko v's Co n s t r u c t ive Co n t in u u m a n d t h e ir A p p li- c a t io n s t o t h e Th e o r y o f Co n s t r u c t ive Fu n c t io n s , Th e L .E .J.B r o u we r Ce n t e n a r y S ym p o - s iu m , N o r t h -H o lla n d P u b l. Co ., A m s t e r d a m , 1 9 8 2 , P p . 2 6 1 { 2 7 3 [2 ] B .A . K u s h n e r , L e c t u r e s o n Co n s t r u c t ive Ma t h e m a t ic a l A n a lys is . ( Tr a n s la t io n fr o m t h e R u s s ia n ) , A MS , P r o vid e n c e , R h o d e Is la n d , 1 9 8 4 7 2