TITLE: Models of computability of partial functions on the reals. ABSTRACT: Six models of computability of partial functions f on the real numbers are studied: two abstract, based on approximable computation w.r.t high level programming languages; two concrete, based on computable tracking functions on the rationals; and two based on polynomial approximation. It is shown that all these models are equivalent, under the assumptions: (1) the domain of f is a union of an effective sequence of rational open intervals, and (2) f is effectively locally uniformly continuous. These conditions are broad enough to includes all the well-known functions of elementary real analysis (rational, exponential, trigonometric, and their inverses, etc.). This work generalises a previously know equivalence result for total functions on the reals.