  A few years ago two of us introduced, motivated by the study of certain
forest-fireprocesses, the self-destructive percolation model (abbreviated as
sdp model). A typical configuration for the sdp model with parameters p and
delta is generated in three steps: First we generate a typical configuration
for the ordinary percolation model with parameter p. Next, we make all sites in
the infinite occupied cluster vacant. Finally, each site that was already
vacant in the beginning or made vacant by the above action, becomes occupied
with probability delta (independent of the other sites).
  Let theta(p, delta) be the probability that some specified vertex belongs, in
the final configuration, to an infinite occupied cluster. In our earlier paper
we stated the conjecture that, for the square lattice and other planar
lattices, the function theta has a discontinuity at points of the form (p_c,
delta), with delta sufficiently small. We also showed remarkable consequences
for the forest-fire models.
  The conjecture naturally raises the question whether the function theta is
continuous outside some region of the above mentioned form. We prove that this
is indeed the case. An important ingredient in our proof is a (somewhat
stronger form of a) recent ingenious RSW-like percolation result of
Bollob\'{a}s and Riordan.
