For a circle
Γ
=
{
z
∈
C
:
|
z
−
c
|
=
ρ
}
\Gamma =\{ z\in \mathbb {C} \colon |z-c|=\rho \}
write
Λ
(
Γ
)
=
{
(
z
,
w
)
:
(
z
−
a
)
(
w
−
a
¯
)
=
ρ
2
,
0
>
|
z
−
a
|
>
ρ
}
\Lambda (\Gamma )=\{ (z,w)\colon \ (z-a)(w-\overline {a}) =\rho ^{2},\ 0>|z-a|>\rho \}
. A continuous function
f
f
on
Γ
\Gamma
extends holomorphically from
Γ
\Gamma
(into the disc bounded by
Γ
\Gamma
) if and only if the function
F
(
z
,
z
¯
)
=
f
(
z
)
F(z,\overline {z})=f(z)
defined on
{
(
z
,
z
¯
)
:
z
∈
Γ
}
\{(z,\overline {z})\colon \ z\in \Gamma \}
has a bounded holomorphic extension into
Λ
(
Γ
)
\Lambda (\Gamma )
. In the paper we consider open connected families of circles
C
\mathcal {C}
, write
U
=
⋃
{
Γ
:
Γ
∈
C
}
U=\bigcup \{ \Gamma \colon \ \Gamma \in \mathcal {C}\}
, and assume that a continuous function on
U
U
extends holomorphically from each
Γ
∈
C
\Gamma \in \mathcal {C}
. We show that this happens if and only if the function
F
(
z
,
z
¯
)
=
f
(
z
)
F(z, \overline {z})=f(z)
defined on
{
(
z
,
z
¯
)
:
z
∈
U
}
\{ (z,\overline {z})\colon z\in U\}
has a bounded holomorphic extension into the domain
⋃
{
Λ
(
Γ
)
:
Γ
∈
Q
}
\bigcup \{ \Lambda (\Gamma )\colon \ \Gamma \in \mathcal {Q}\}
for each open family
Q
\mathcal {Q}
compactly contained in
C
\mathcal {C}
. This allows us to use known facts from several complex variables. In particular, we use the edge of the wedge theorem to prove a theorem on real analyticity of such functions.