Continuity Theorems for a Class of Computable Operators

Computable operators corresponding to the concept of a left computably enumerable real number, called $$\mathrm{L}$$ -operators, are studied. Their continuity properties most commonly used in the constructive mathematical analysis of A. A. Markov’s school are examined. It is proved that any $$\mathrm{L}$$ -operator is nondecreasing and almost left continuous. An example of an $$\mathrm{L}$$ -operator which is neither left continuous nor right pseudocontinuous at some point is constructed. An almost continuity criterion for an $$\mathrm{L}$$ -operator is found. This criterion is used to prove that almost continuous $$\mathrm{L}$$ -operators are not necessarily continuous or pseudouniformly continuous on a closed interval.