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Let $D^{\text{'}}\subset {ℂ}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\text{'}},z_n)$ a holomorphic function in the cylinder $D=D^{\text{'}}\times U_n$ and continuous on $\overline{D}$. If for each fixed $a^{\text{'}}$ in some set $E\subset \partial D^{\text{'}}$, with positive Lebesgue measure $\text{mes}E>0$, the function $f(a^{\text{'}},z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\text{'}},z_n)$ can be holomorphically continued to $(D^{\text{'}}\times ℂ)\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\text{'}}\times ℂ$.