Let
B
\mathbb {B}
be the open unit ball in
C
2
\mathbb {C}^2
and let
a
,
b
,
c
a, b, c
be three points in
C
2
\mathbb {C}^2
which do not lie in a complex line, such that the complex line through
a
,
b
a, b
meets
B
\mathbb {B}
and such that if one of the points
a
,
b
a, b
is in
B
\mathbb {B}
and the other in
C
2
∖
B
¯
\mathbb {C}^2\setminus \overline {\mathbb {B}}
then
⟨
a
|
b
⟩
≠
1
\langle a|b\rangle \not = 1
and such that at least one of the numbers
⟨
a
|
c
⟩
,
⟨
b
|
c
⟩
\langle a|c\rangle ,\ \langle b|c\rangle
is different from
1
1
. We prove that if a continuous function
f
f
on
b
B
b\mathbb {B}
extends holomorphically into
B
\mathbb {B}
along each complex line which meets
{
a
,
b
,
c
}
\{ a, b, c\}
, then
f
f
extends holomorphically through
B
\mathbb {B}
. This generalizes the recent result of L. Baracco who proved such a result in the case when the points
a
,
b
,
c
a, b, c
are contained in
B
\mathbb {B}
. The proof is quite different from the one of Baracco and uses the following one-variable result, which we also prove in the paper: Let
Δ
\Delta
be the open unit disc in
C
\mathbb {C}
. Given
α
∈
Δ
\alpha \in \Delta
let
C
α
\mathcal {C}_\alpha
be the family of all circles in
Δ
\Delta
obtained as the images of circles centered at the origin under an automorphism of
Δ
\Delta
that maps
0
0
to
α
\alpha
. Given
α
,
β
∈
Δ
,
α
≠
β
\alpha , \beta \in \Delta ,\ \alpha \not = \beta
, and
n
∈
N
n\in \mathbb {N}
, a continuous function
f
f
on
Δ
¯
\overline {\Delta }
extends meromorphically from every circle
Γ
∈
C
α
∪
C
β
\Gamma \in \mathcal {C}_\alpha \cup \mathcal {C}_\beta
through the disc bounded by
Γ
\Gamma
with the only pole at the center of
Γ
\Gamma
of degree not exceeding
n
n
if and only if
f
f
is of the form
f
(
z
)
=
a
0
(
z
)
+
a
1
(
z
)
z
¯
+
⋯
+
a
n
(
z
)
z
¯
n
(
z
∈
Δ
)
f(z) = a_0(z)+a_1(z)\overline z +\cdots +a_n(z)\overline z^n (z\in \Delta )
where the functions
a
j
,
0
≤
j
≤
n
a_j, 0\leq j\leq n
, are holomorphic on
Δ
\Delta
.