Let
E
E
be a subset of the unit circumference
C
C
. If for every nonempty open arc
A
A
of
C
C
, the set
E
E
is not both metrically dense and of second category in
A
A
, then there exists a nonconstant analytic function
f
f
on the open unit disk
Δ
\Delta
, such that
f
∗
(
η
)
=
0
{f^ * }(\eta ) = 0
,
η
∈
E
\eta \in E
, where
f
∗
{f^ * }
is the radial limit function of
f
f
.