A range of values of a real function $f$ : $E^{d} $\rightarrow \Re$ can be used to implicitly define a subset of Euclidean space $E^{d}$. Such `implicit functions' have many uses in geometric and solid modeling. This paper focuses on the properties and construction of real functions for the representation of rigid solids (compact, semi-analytic, and regular subsets of $E^{d}$). We review some known facts about real functions defining compact semi-analytic sets, and their applications. The theory of $R$-functionsdeveloped in [Rva82] provides means for constructing real function representations of solids described by the standard (non-regularized) set operations. But solids are not closed under the standard set operations and such representations are rarely available in modern modeling systems. More generally, assuring that a real function $f$ represents a regular set may be difficult. Until now, the regularity has either been assumed, or treated in an ad hoc fashion. We show that topological and extremal properties of real functions can be used to test for regularity, and discuss procedures for constructing real functions with desired properties for arbitrary solids.