  The concept of Hausdorff continuous interval valued functions, developed
within the theory of Hausdorff approximations and originaly defined for
interval valued functions of one real variable is extended to interval valued
functions defined on a topological space X. The main result is that the set of
all finite Hausdorff continuous functions on any topological space X is
Dedekind order complete. Hence it contains the Dedekind order completion of the
set C(X) of all continuous real functions defined on X as well as the Dedekind
order completion of the set C_b(X) of all bounded continuous functions on X.
Under some general assumptions about the topological space X the Dedekind order
completions of both C(X) and C_b(X) are characterised as subsets of the set of
all Hausdorff continuous functions. This solves a long outstanding open problem
about the Dedekind order completion of C(X). In addition, it has major
applications to the regularity of solutions of large classes of nonlinear PDEs.
