The language LRTp is a non-deterministic language for exact real number computation. It has been shown that all computable first order relations in the sense of Brattka are definable in the language. If we restrict the language to single-valued total relations (e.g. functions), all polynomials are definable in the language. This paper is an expanded version of [12] in which we show that the non-deterministic version of the limit operator, which allows to define all computable first order relations, when restricted to single-valued total inputs, produces single-valued total outputs. This implies that not only the polynomials are definable in the language but also all computable first order functions.