We prove that in wide generality the critical curve of the activated random
walk model is a continuous function of the deactivation rate, and we provide a
bound on its slope which is uniform with respect to the choice of the graph.
Moreover, we derive strict monotonicity properties for the probability of a
wide class of `increasing’ events,extending previous results of Rolla and
Sidoravicius (2012). Our proof method is of independent interest and can be
viewed as a reformulation of the `essential enhancements’ technique — which
was introduced for percolation — in the framework of Abelian networks.