Electroencephalography (EEG) signals convey information related to different processes that take place in the brain. From the EEG fluctuations during sleep, it is possible to establish the sleep stages and identify short events, commonly related to a specific physiological process or pathology. Some of these short events (called A-phases) present an organization and build up the concept of the Cyclic Alternating Pattern (CAP) phenomenon. In general, the A-phases abruptly modify the EEG fluctuations, and a singular behavior could occur. With the aim to quantify the abrupt changes during A-phases, in this work the wavelet analysis is considered to compute Hölder exponents, which measure the singularity strength. We considered time windows of 2s outside and 5s inside A-phases onset (or offset). A total number of 5121 A-phases from 9 healthy participants and 10 patients with periodic leg movements were analyzed. Within an A-phase the Hölder numerical value tends to be 0.6, which implies a less abrupt singularity. Whereas outside of A-phases, it is observed that the Hölder value is approximately equal to 0.3, which implies stronger singularities, i.e., a more evident discontinuity in the signal behavior. In addition, it seems that the number of singularities increases inside of A-phases. The numerical results suggest that the EEG naturally conveys singularities modified by the A-phase occurrence, and this information could help to conceptualize the CAP phenomenon from a new perspective based on the sharpness of the EEG instead of the oscillatory way.