  Context. Blazar flares provide a window into the extreme physical processes
occurring in relativistic outflows. Most numerical codes used for modeling
blazar emission during flares utilize a simplified continuous-loss description
of particle cooling due to the inverse Compton (IC) process, neglecting
non-continuous (discrete) effects that arise in the Klein-Nishina (KN) regime.
The significance of such effects has not been explored in detail yet. Aims. In
this study, we investigate the importance of non-continuous Compton cooling
losses and their impact on the electron spectrum and spectral energy
distribution (SED) of blazars during high flux states (flares), as well as in
the low state. Methods. We solve numerically the full transport equation
accounting for large relative jumps in energy, by extending our existing blazar
flare modeling code EMBLEM. We perform a detailed physical modeling of the
brightest gamma-ray flare of the archetypal Flat Spectrum Radio Quasar (FSRQ)
3C 279 detected in June 2015. We then compare results obtained using the full
cooling term and using the continuous-loss approximation. Results. We show that
during flaring states of FSRQs characterized by high Compton dominance, the
non-continuous cooling can lead to a significant modification of the electron
spectrum, introducing a range of distinct features, such as low-energy tails,
hardening/softening, narrow and extended particle excesses, and shifts in the
cooling break position. Such distortion translates to differences in the
associated SED up to 50%. This highlights the importance of non-continuous
effects and the need to consider them in blazar emission models, particularly
applied to extreme gamma-ray flares.
