In this paper we discuss the algebraic structure of the space H(X) of finite Hausdorff continuous interval functions defined on an arbitrary topological space X.  In particular, we show that H(X) is a linear space over R containing C(X), the space of continuous real functions on X, as a linear subspace.  In addition, we prove that the order on H(X) is compatible with the linear structure introduced here so that H(X) is an Archimedean vector lattice