A SIMPLE CHARACTERIZATION OF THE COMPUTABILITY OF REAL FUNCTIONS

The TTE-approach to computability of real functions uses infinitary names of the argument's and the function's values, computability being defined as the existence of some algorithmic procedure transforming the names of any argument's value into ones of the corresponding value of the function. Two ways to avoid using such names are considered in the present paper. At each of them, the corresponding characterization of computability of real functions is through the existence of an appropriate recursively enumerable set establishing some relation between rational approximations of the argument's value and rational approximations of the corresponding value of the function.
The characterizations in question are derived from ones for computability of functions in metric and in topological spaces.