There might be a temptation to change the comparator to an op amp. This is poor practice for any circuit which requires high speed performance because the op amp output would saturate causing uncontrolled time delays. The comparator is subject to multiple switching caused by high frequency noise riding on slow rising input signals. The multiple switching is eliminated by the capacitor, C, which is placed across its input leads. Effectively, C couples the high frequency noise onto both comparator leads enabling it to be rejected by the comparator's common-mode rejection capability. Notice in the waveform that there is no multiple switching taking place.