Is the eyes' movement discrete?
There are two types of eye movement: smooth pursuit and saccadic. As the name suggests the latter movement involves quick and discontinuous movements of the eyes. Saccadic movement is used most of the time as the eyes move around analysing the current scene.

According to Wikipedia, these saccades are the fastest movements produced by the human body: peak angular speed of 900 degrees of rotation per second (this would be 2.5 full rotations of the eyeball per second, were such a thing possible).

The processing of information by the eye-brain system during a saccade is complex, and may be responsible for the effect that you report: it causes the phenomenon of saccadic masking explained in this quote from Wikipedia:


  A person may observe the saccadic masking effect by standing in front of a mirror and looking from one eye to the next (and vice versa). The subject will not experience any movement of the eyes nor any evidence that the optic nerve has momentarily ceased transmitting. Due to saccadic masking, the eye/brain system not only hides the eye movements from the individual but also hides the evidence that anything has been hidden. Of course, a second observer watching the experiment will see the subject's eyes moving back and forth. The function's main purpose is to prevent smearing of the image.