Let D ' ⊂ Cn−1 be a bounded domain of Lyapunov and f(z ' , zn) a holomorphic function in the cylinder D = D' × Un and continuous on D. If for each fixed a 0 in some set E ⊂ ∂D' , with positive Lebesgue measure mes E > 0, the function f(a ' , zn) of zn can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then f(z ' , zn) can be holomorphically continued to (D ' × C) \ S, where S is some analytic (closed pluripolar) subset of D ' × C.