Continuity properties of constructive operators on the computably enumerable reals
0000-0002-4519-9696 This paper studies the almost continuity property (which is one of the effective modifications of the classical continuity property in constructive mathematical analysis of the A. A. Markov school) of constructive operators defined on left-computably enumerable (left-c.e.) reals, called $\ML$-operators. A criterion is given for the almost continuity of an arbitrary $\ML$-operator $F$ increasing on the segment $[a,b]$, which consists in the fact that for any real $\varepsilon>0$ and for any real $\alpha$ belonging to the image of the segment $[a,b]$ with respect to $F$, there exists a real $\beta$ belonging to the same image and satisfying the condition $\alpha<\beta<\alpha+\varepsilon$. The presented criterion is compared with the well-known continuity criterion (fr om the standpoint of classical mathematical analysis) of an increasing function $f$ on the segment $[a,b]$, according to which $f$ is continuous on $[a,b]$ if and only if any real belonging to the segment $[f(a),f(b)]$ is a value of $f$. The existence of an increasing almost continuous $\ML$-operator is proved, whose range consists of all left-c.e. reals that do not belong to a given interval $[c,d]$, wh ere $c\leqslant d$ are arbitrary rationals. An increasing $\ML$-operator is constructed that is not pseudocontinuous at~$0$. An increasing almost continuous $\ML$-operator $F$ is also constructed, which does not have an inverse $\ML$-operator on the segment $[0,1]$ and that is not continuous on this segment, such that each left-c.e. real from the segment $[F(0),F(1)]$ is its value
УДК 510.57+510.25
${file_?????}Keywords: left-c.e. real, constructive operator, almost continuous operator, pseudocontinuous operator, increasing operator.
_______