We investigate the relationships between the infinitely many
characteristic zeros (or modes) of linear systems
subject to point delays and their delay-free counterparts based
on algebraic results and theory of analytic functions.
The cases when the delay tends to zero or to infinity are
emphasized in the study. It is found that when the delay is
arbitrarily small, infinitely many of those zeros are located in
the stable region with arbitrarily large modulus, while their
contribution to the system dynamics becomes irrelevant. The
remaining finite characteristic zeros converge to those of the
delay-free nominal system. When the delay tends to infinity,
infinitely many zeros are close to the origin. Furthermore, there
exist two auxiliary delay-free systems which describe the
relevant dynamics in both cases for zero and infinite delays. The
maintenance of the delay-free system stability in the presence of
sufficiently small delayed dynamics is also discussed in
light of $H_\infty$ -theory. The main mathematical
arguments used to derive the results are based on the theory of
analytic functions.