The language is used to express many algorithms in scientific computing where `while' programs are applied to continuous data. In the theory of data, continuous data types are modelled by topological many-sorted algebras. We study both exact and approximate computations by `while' programs, and `while' programs with arrays, over topological many-sorted algebras with partial operations. First, we establish that partial operations are necessary in order to compute a wide range of continuous functions. We prove basic continuity properties of our abstract computability:

Any partial function computable over a partial topological algebra by a `while'-array program is continuous. Any set semicomputable, or computable, over a partial topological algebra by a `while'-array program is open, or clopen, respectively.

Secondly, we contrast exact and approximate computations.
The class of functions that can be computed exactly can be
quite limited. We show that on connected total algebras, the
`while' and `while'-array computable functions are
precisely those that are explicitly definable by terms.
We show that for certain general classes of topological
algebras, the total functions that can be approximated by
`while' programs are precisely those that can be
effectively approximated by terms. This property we call
generalised Weierstrass approximation. An application of
this result is that a function on the set **R** of reals is
computable in the sense of computable analysis if, and only
if, it is `while' approximable on a simple algebra based on **R**.