small bandwidth of comparator
A comparator is a high gain amplifier with hysteresis. Even very small voltage difference at the input should make comparator (amplifier) output high (or low)saturation. For this reason, amplifier gain should be high.
Additionally, amplifiers has certain bandwith. The gain is high in the bandwitdh frequencies only. For higher frequencies the amplifier gain decrease and so, comparator doesn't work properly and doesn't operate as a comparator.