  We study the critical behavior of the one-dimensional random field Ising
model (RFIM) with long-range interactions ($\propto r^{-(d+\sigma)}$) by the
nonperturbative functional renormalization group. We find two distinct regimes
of critical behavior as a function of $\sigma$, separated by a critical value
$\sigma_c$. What distinguishes these two regimes is the presence or not of a
cusp-like nonanalyticity in the functional dependence of the renormalized
cumulants of the random field at the fixed point. This change of behavior can
be associated to the characteristics of the large-scale avalanches present in
the system at zero temperature. We propose ways to check these predictions
through lattice simulations. We also discuss the difference with the RFIM on
the Dyson hierarchical lattice.
