In this paper we prove a result which, apart from having independent interest, has found applications in recent mathematical economic literature of rational choice theory. The result states that if a two-dimensional demand function satisfies budget exhaustion, the Weak Axiom of Revealed Preference and its range contains the strictly positive orthant of two dimensional Euclidean space, then it is representable by an utility function which is upper semicontinuous on the non-negative orthant of two dimensional Euclidean space and strictly quasi-concave and strictly monotonically increasing on the strictly positive orthant of two dimensional Euclidean space. By strictly monotonically increasing on the strictly positive orthant of two dimensional Euclidean space we mean that if a strictly positive vector is semi-strictly greater than another vector in the non-negative orthant of two dimensional Euclidean space, then the former has greater utility than the latter.