  Let $E$ and $F$ be Banach lattices. We show first that the disjointness
preserving linear functionals separate the points of any infinite dimensional
Banach lattice $E$, which shows that in this case the unbounded disjointness
operators from $E\to F$ separate the points of $E$. Then we show that every
disjointness preserving operator $T:E\to F$ is norm bounded on an order dense
ideal. In case $E$ has order continuous norm, this implies that that every
unbounded disjointness preserving map $T:E\to F$ has a unique decomposition
$T=R+S$, where $R$ is a bounded disjointness preserving operator and $S$ is an
unbounded disjointness preserving operator, which is zero on a norm dense
ideal. For the case that $E=C(X)$, with $X$ a compact Hausdorff space, we show
that every disjointness preserving operator $T:C(X)\to F$ is norm bounded on an
norm dense sublattice algebra of $C(X)$, which leads then to a decomposition of
$T$ into a bounded disjointness operator and a finite sum of unbounded
disjointness preserving operators, which are zero on order dense ideals.
