Let
S
S
be a compact Hausdorff space and
X
X
a complex manifold. We consider the space
C
(
S
,
X
)
C(S,X)
of continuous maps
S
→
X
S\to X
, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space
B
(
S
,
X
)
B(S,X)
of Borel maps. As an application we prove two theorems about bounded holomorphic functions on
C
(
S
,
X
)
C(S,X)
, one reminiscent of the Monodromy Theorem, the other of Liouville’s Theorem.