How does the strict phenomenologist deal with atypical forms of consciousness that an analytical philosopher need only point to brain function to explain? For instance, how does phenomenology deal with the problem of sleeping, because discontinuity of awareness seems problematic?
Phenomenologists redefine consciousness to include the past, present, and future as subject to its inspection. Therefore, sleep, which may lead to an interruption of perception doesn't lead to an interruption of consciousness per se.
A good introduction to phenomenology and time-consciousness can be found at the Internet Encyclopedia of Philosophy. Edmund Husserl is considered the founder of phenomenology, but there are many later philosophers in the tradition such as Martin Heidegger, Jean-Paul Sartre, Maurice Merleau-Ponty, and Jacques Derrida,
The question you have asked is an excellent question. Husserl who considered awareness an "intentional" activity had similar concerns:
[T]ime-consciousness underscores... other intentional acts because these other intentional acts presuppose or include the consciousness of internal time. For this and other reasons, Husserl, in his On the Phenomenology of the Consciousness of Internal Time (1893-1917) (1991), deemed time-consciousness the most “important and difficult of all phenomenological problems”
The short version is he rejected the idea of Newtonian time as fundamental, that is, to define consciousness in such a way that it needn't account for a sequence of discrete moments in the classical physicalist sense, but rather extend consciousness over the past, present, and future. Hence, for a phenomenologist, consciousness is broadened to include past in order to create a unity of temporal experience. From the article:
Husserl attempts to argue that consciousness extends to capture past moments of experience and temporal objects therein by “retaining” and “protending” the elapsed and yet to come phases of its experience and thereby the past words that do not presently exist.