We consider two classical computability notions for functions mapping all computable real numbers to computable real numbers. It is clear that any function that is computable in the sense of Markov, i.e., computable with respect to a standard Gödel numbering of the computable real numbers, is computable in the sense of Banach and Mazur, i.e., it maps any computable sequence of real numbers to a computable sequence of real numbers. We show that the converse is not true. This solves a (...) long-standing open problem posed by Kushner. (shrink)

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the (...) theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well-known non-appropriateness of the representation to some base b by proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers. (shrink)