  We point out that in the first order time-dependent perturbation theory, the
transition probability may behave nonsmoothly in time and have kinks
periodically. Moreover, the detailed temporal evolution can be sensitive to the
exact locations of the eigenvalues in the continuum spectrum, in contrast to
coarse-graining ideas. Underlying this nonsmooth and level-resolved dynamics is
a simple equality about the sinc function $\sinc x \equiv \sin x / x$. These
physical effects appear in many systems with approximately equally spaced
spectra, and is also robust for larger-amplitude coupling beyond the domain of
perturbation theory. We use a one-dimensional periodically driven tight-binding
model to illustrate these effects, both within and outside the perturbative
regime.
