When reading a story, the reader requires additional information to be connected in some way with information that came prior — if it is a continuous narrative.
With a discontinuous narrative, the reader will be looking for ways to fit the fragments of information together like a jigsaw puzzle.
Elsewise, they might as well be touring a museum of modern art.
Now, there is some confusion as to the function of ‘names’. Briefly, most names as we know and use them are simply abbreviated handles, or labels, for the purposes of swift and easy identification.
As some other answers noted, if your story was done from POV of the unnamed character, then there is less need for name — but that doesn't mean that it can't occur.
Are you in need of a recognizable, succinct, and semantically portable noun whereby to identify a personage to your readers?
A narration can continue with the absence of anything so long as that thing is not a necessary component thereof: if there is a place where something should occur, and it doesn't, then readers will notice.
“Is this person afraid to utter that name?”
“Did the writer forget that the main character hasn't been given a name yet?”
Once you do give something, it follows that such revelation should be connected with previous parts of the narration.
How exactly to maintain a chain in a narration, and when it is possible to sever or fragment that chain, depends on many things:
- the intelligence, attentiveness, and recall of the reader
- the intelligibility of the prose — if it is confusing and muddled, or otherwise difficult to read
- the rewards gained by delving into the prose — if a reader sees evidence of profound investment in the writing, i.e. gaining insights by contemplating even terse or obscure writs, then they could be annoyed and disinterested but they won't think the writer inept or a hack
A ‘name’ as a part of the narrated events is exactly that; do not introduce them in any way that seems inexcusably contrived unless such is purposefully required — i.e. it only seemingly ex machina. If so, however, keep in mind that you ought be at a certain level of trust with your readers.
A ‘name’ used to conveniently identify a character or other personage to your readers serves a narrative function. Any such devices are usually referred to as, well, narrative devices.
The exact definition and optimal implementation of a narrative device varies, but they typically can be grouped into two original classes:
- voice of the narrator
This should be intuitive enough: Once upon a time, there was a certain woman. Let us refer to her as Alice, for that is — suitably enough — her given name.
The narrator could be one of the persons featured in the story, it may be a layer that exists between the story and the reader, or it may be the author. It could be personified, or it could simply be a certain attitude and vocabulary conveyed by the choice of words and grammar.
- premise or foundation of the story
If something is given as a precept or axiom, not explored narratively, or otherwise taken for granted. It could be that you want to explore some strange and bizarre medical condition, and how it compares to so–called “normal life”, and the triumph of Love or whatever, but you don't really want to explore the clinical diagnoses or origins of the condition itself.