You start off battery, you run off alternator, the alternator recharges the battery. If the battery terminals are corroded, even if the alternator is putting out like it should, the battery will not be able to accept a charge. I suspect you need a battery, and that your connectors are "terminally" ill. Usually, when an alternator stops charging like it should, your lights will start coming on in the dash. Either all of the little red lights, or the SRS light, etc. You generally get a warning. Your battery, I believe, was giving you a warning, and yes...they can just crap out suddenly. Usually the first really cold day, or the first really hot day.

Chris