  The goal of this paper is to present a certain generalization of the
classical Kontinuit\"atssatz of Behnke for holomorphic/meromorphic functions in
terms of the lift to the envelope of holomorphy. We consider two non-equivalent
formulations: "discrete" and "continuous" ones. Giving a proof of the
"discrete" version we, somehow unexpectedly, construct a counterexample to the
"continuous" one when convergence/continuity of analytic sets is considered in
Hausdorff topology or, even in the stronger topology of currents. But we prove
the "continuous" version of the Kontinuit\"atssatz if continuity is understood
with respect to the Gromov topology. Our formulations seem to be not yet
existing in the literature. A number of relevant examples and open questions is
given as well.

Complex Variables