  The rational Dunkl operators are commuting differential-reflection operators
on the Euclidean space $\RR^d$ associated with a root system. The aim of the
paper is to study local boundary behaviour of generalized harmonic functions
associated with the Dunkl operators. We introduce a Lusin-type area integral
operator $S$ by means of Dunkl's generalized translation and the Dunkl
operators. The main results are on characterizations of local existence of
non-tangential boundary limits of a generalized harmonic function $u$ in the
upper half-space $\RR^{d+1}_+$ associated with the Dunkl operators, and for a
subset $E$ of $\RR^d$ invariant under the reflection group generated by the
root system, the equivalence of the following three assertions are proved: (i)
$u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is
non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)$ is finite for a.e.
$x\in E$.
