[en] The escape of particles from the phase space produced by a two-dimensional, nonlinear and area-preserving, discontinuous map is investigated by using both numerical simulations and the explicit solution of the corresponding diffusion equation. The mapping, given in action-angle variables, is parameterized by K, which controls a transition from integrability to non-integrability. We focus on the two dynamical regimes of the map: slow diffusion ($K<<!--\; <\; -->K_c$) and quasilinear diffusion ($K><!--\; >\; -->K_c$) regimes, separated by the critical parameter value K_c = 1. When a hole is introduced in the action axis, we find the histogram of escape times $P_{\mathrm{E}}(n)$ and the survival probability of particles $P_{\mathrm{S}}(n)$ to be scaling invariant in both the slow and the quasilinear diffusion regimes, with scaling laws proportional to the corresponding diffusion coefficients, namely, proportional to $K^{5/2}$ and K², respectively. Our numerical simulations agree remarkably well with the analytical results obtained from the explicit solution of the diffusion equation, hence giving robustness to the escape formalism. (paper)