Continuity Properties of Constructive Operators on the Set ofComputably Enumerable Real Numbers

In the present article, we study almost continuous operators on the set of left computably enumerable (left-c.e.) real numbers (called L -operators). In constructive analysis developed in Markov’s school, almost continuity is one of effective modifications of the classical continuity. We suggest a criterion for almost continuity of an L -operator F that increases on a segment $$ [a,b] $$ ; namely, for every positive real ε and every number α in the image of $$ [a,b] $$ under F , there exists a number β in the same image such that α <β <α +ε . We compare this criterion with the well-known criterion from classical calculus for an increasing function f on a segment $$ [a,b] $$ to be continuous; namely, every real in the segment $$ [f(a),f(b)] $$ belongs to the range of f . We show that there exists an increasing almost continuous L -operator whose range is the set of all left-c.e. reals that do not belong to a prescribed interval $$ [c,d] $$ with rational endpoints and c d . We construct an increasing L -operator that is not pseudocontinuous at $$ 0 $$ . We also construct an increasing almost continuous L -operator F such that F admits no inverse L -operator on $$ [0,1] $$ and is not continuous on this segment but each left-c.e. real in $$ [F(0),F(1)] $$ belongs to the range of F .