Let
X
X
be a space of homogeneous type and
W
W
a subset of
X
×
(
0
,
∞
)
X \times (0,\infty )
. Then, under minimal conditions on
W
W
, we obtain a relationship between two modes of convergence at the boundary
X
X
for functions defined on
W
W
. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.