### Abstract

>The aim of this thesis is to contribute to close the gap existing
between the theory of computable analysis and actual computation. In order
to study computability over real numbers we use several tools peculiar
to the theory of programming languages. In particular we introduce a special
kind of typed lambda calculus as an appropriate formalism for describing
computations on real numbers. Furthermore we use domain theory, to give
semantics to this typed lambda calculus and as a conseguence to give a
notion of computability on real numbers. We discuss the adequacy of Scott-Domains
as domains for representing real numbers. We relate the Scott topology
on such domains to the euclidean topology on R.
Domain theory turns out to be useful also in the study of higher order
functions. In particular one of the most important results contained in
this thesis concerns the characterisation of the topological properties
of the computable higher order functions on reals. Our approach allows
moreover to phrase and discuss w.r.t. real numbers issues of programming
languages. We address the problem of defining an implementation of real
numbers as an abstract data type. Finally we investigate algorithms for
carrying out efficient computations on reals.