  Several approaches to the dynamics of loop quantum gravity involve
discretizing the equations of motion. The resulting discrete theories are known
to be problematic since the first class algebra of constraints of the continuum
theory becomes second class upon discretization. If one treats the second class
constraints properly, the resulting theories have very different dynamics and
number of degrees of freedom than those of the continuum theory. It is
therefore questionable how these theories could be considered a starting point
for quantization and the definition of a continuum theory through a continuum
limit. We show explicitly in a model that the {\em uniform discretizations}
approach to the quantization of constrained systems overcomes these
difficulties. We consider here a simple diffeomorphism invariant one
dimensional model and complete the quantization using {\em uniform
discretizations}. The model can be viewed as a spherically symmetric reduction
of the well known Husain--Kucha\v{r} model of diffeomorphism invariant theory.
We show that the correct quantum continuum limit can be satisfactorily
constructed for this model. This opens the possibility of treating 1+1
dimensional dynamical situations of great interest in quantum gravity taking
into account the full dynamics of the theory and preserving the space-time
covariance at a quantum level.
