  Let $f$ be a holomorphic, or even meromorphic, function on the unit disc.
Plessner's theorem then says that, for almost every boundary point $\zeta $,
either (i) $f$ has a finite nontangential limit at $\zeta $, or (ii) the image
$f(S)$ of any Stolz angle $S$ at $\zeta $ is dense in the complex plane. This
paper shows that statement (ii) can be replaced by a much stronger assertion.
This new theorem and its analogue for harmonic functions on halfspaces also
strengthen classical results of Spencer, Stein and Carleson.
