  The dynamics of physical theories is usually described by differential
equations. Difference equations then appear mainly as an approximation which
can be used for a numerical analysis. As such, they have to fulfill certain
conditions to ensure that the numerical solutions can reliably be used as
approximations to solutions of the differential equation. There are, however,
also systems where a difference equation is deemed to be fundamental, mainly in
the context of quantum gravity. Since difference equations in general are
harder to solve analytically than differential equations, it can be helpful to
introduce an approximating differential equation as a continuum approximation.
In this paper implications of this change in view point are analyzed to derive
the conditions that the difference equation should satisfy. The difference
equation in such a situation cannot be chosen freely but must be derived from a
fundamental theory. Thus, the conditions for a discrete formulation can be
translated into conditions for acceptable quantizations. In the main example,
loop quantum cosmology, we show that the conditions are restrictive and serve
as a selection criterion among possible quantization choices.
