  We consider the equation $$ \dot x(t)=f(t,x(t),x(\eta(t))) $$ with a variable
time-shift $\eta(t)$. Both the nonlinearity $f$ and the shift function $\eta$
are given, and are assumed to be analytic (that is, holomorphic) functions of
their arguments. Typically the time-shift represents a delay, namely that
$\eta(t)=t-r(t)$ with $r(t)\ge 0$. The main problem considered is to determine
when solutions (generally $C^\infty$ and often periodic solutions) of the
differential equation are analytic functions of $t$; and more precisely, to
determine for a given solution at which values of $t$ it is analytic, and at
which values it is not analytic. Both sufficient conditions for analyticity,
and also for nonanalyticity, at certain values of $t$ are obtained. It is shown
that for some equations there exists a solution which is $C^\infty$ everywhere,
and is analytic at certain values of $t$ but is not analytic at other values of
$t$. Throughout our analysis, the dynamic properties of the map $t\to \eta(t)$
play a crucial role.
