Let **R**^{*}:= **R** **u*** {-infty, +infty}*^{7} denote the real numbers adjoined
with *-infty* and *+infty.* A subset *S *__c__* R^{*}* is a

*forall x***e****R**^{*}exist s**e***S: s*__<__x /\ forall s'**e***S: s'*and__<__x => s'__<__s*forall x***e****R**^{*}exist s**e***S: s*__>__x /\ forall s'**e***S: s'*__>__x => s'__>__s,

and

S_{b,l}:= { 0 }u{ x = * m ·b^{e}|*e{+,-}, eeZ,m=SUM_{i=1}^{l}x[i] b^{-i},x[i]e{0,1, ..., b-1}, x[1] =/= 0 }

whereS(b,l,e1,e2) := { xeS_{b,l}| e1<e<e2 },

*b***e****N**, b > 1,*base*,*l***e****N**,*length of the mantissa*,*e1, e2***e****Z**,*bounds of the exponent*.

S(b,l)^{*}:= S(b,l)u{-infty, +infty}, andS(b,l,e1,e2)^{*}:= S(b,l,e1,e2)u{-infty, +infty}.

A *rounding* is a map *r: R^{*} -> S* to a screen

We call itforall seS: r(s) =s.

For example the mappingsforall x,yeR^{*}: x<y => r(x)<r(y).

wherephi: {

R^{*}-> Sx^{|}->|_|{ seS | s<x }and psi: {

R^{*}-> Sx^{|}->|¯|{ seS | s>x },

CoFI Note: M-7 -- Version: 0.2 -- 13 April 1999.

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