The main goal of this paper is to prove the following statement. Let f = ∑^n∈E a_n^z be a function holomorphic and of bounded characteristic in the unit disk D, where E is a ∧(1)-subset of ℤ₊. Assume that f has a pseudocontinuation of bounded characteristic to the annulus {z ∈ ℂ : 1 < |z| < R}. Then f admits analytic continuation to the disk RD. In particular, f is a polynomial if R = +∞. Bibliography: 16 titles.