  Although the widely-used analytical afterglow model of gamma-ray bursts
(GRBs) predicts a sharp cooling break $\nu_c$ in its afterglow spectrum, the
GRB observations so far rarely show clear evidence for a cooling break in their
spectra or its corresponding temporal break in their light curves. Employing a
Lagrangian description of the blast wave, we conduct a sophisticated
calculation of the afterglow emission. We precisely follow the cooling history
of non-thermal electrons accelerated into each Lagrangian shell. We show that a
detailed calculation of afterglow spectra does not in fact give rise to a sharp
cooling break at $\nu_c$. Instead, it displays a very mild and smooth
transition, which occurs gradually over a few orders of magnitude in energy or
frequency. The main source of this slow transition is that different
mini-shells have different evolution histories of the comoving magnetic field
strength $B$, so that deriving the current value of $\nu_c$ of each mini-shell
requires an integration of its cooling rate over the time elapsed since its
creation. We present the time evolution of optical and X-ray spectral indices
to demonstrate the slow transition of spectral regimes, and discuss the
implications of our result in interpreting GRB afterglow data.
