A new approach to computability on realnumbers

We introduce majorant computability of functions on reals. A structural theorem is proved, which connects the graph of a majorant{computable function with validity of a nite formula on the set of the hereditarily nite set HF( R) (where R is an elementary proper extension of standard real numbers). The class of majorant{computable functions in our approach include an interesting class of real total functions having meromorphic extension on C. This class in particular contains functions which are solutions of known di erential equations. The notion of a majorant{computable functional on the set of total majorant{computable real functions is de ned. As an example of a majorant{computable functional the Riemann integral is proposed. 1 Section