  Given a pseudoconvex domain D in C^N, N>1, we prove that there is a
holomorphic function f on D such that the lengths of paths p: [0,1]--> D along
which Re f is bounded above, with p(0) fixed, grow arbitrarily fast as p(1)-->
bD. A consequence is the existence of a complete closed complex hypersurface M
in D such that the lengths of paths p:[0,1]--> M, with p(0) fixed, grow
arbitrarily fast as p(1)-->bD.
