While delay differential equations with variable delays may have a superficial appearance of analyticity, it is far from clear in general that a global bounded solution $x(t)$ (namely, a bounded solution defined for all time $t$, such as a solution lying on an attractor) is an analytic function of $t$. Indeed, very often such solutions are not analytic, although they are often $C^\infty$. In this talk we provide sufficient conditions both for analyticity and for non-analyticity (but $C^\infty$ smoothness) of such solutions. In fact these conditions may occur simultaneously for the same solution, but in different regions of its domain, and so the solution exhibits co-existence of analyticity and non-analyticity. In fact, we show it can happen that the set of non-analytic points $t$ of a solution $x(t)$ can be a generalized Cantor set.