### Abstract

We present the different constructive definitions of real number that can
be found in the literature. Using domain theory
we analyse the notion of computability that is substantiated by these
definitions and we give a definition of computability for real numbers and
for functions acting on them.
This definition of computability turns out to be equivalent to
other definitions given in the literature using different
methods.
Domain theory is a useful tool to study higher order computability on
real numbers. An interesting connection between Scott-topology and the
standard topologies on the real line and on the space of continuous
functions on reals is stated. The main original result in this paper is
the proof that every computable functional on real numbers is continuous
w.r.t. the compact open topology on the function space.