  In this paper we check the validity of the ideas of double scaling as given
by Ball and Forte in a semi inclusive process like $J/\psi$ production at HERA,
in different kinematical regions, for low values of the Bjorken variable $x$.
In particular, we study $J/\psi$ production in the inelastic and diffractive
(elastic) regimes using the double scaling form of the gluon distribution
functions. We compare these predictions with data (wherever available) and with
other standard parameterisations. We find that double scaling holds in the
inelastic regime over a larger kinematic region than that given by the analysis
of the proton structure function $F_2^p$. However, in the diffractive region,
double scaling seems to suggest an admixture of hard pomeron boundary
conditions for the gluon distribution, while predicting a steeper rise in the
cross section than that suggested by present data.
