How much time out of the US breaks continuous residence? Why?

Generally a single absence of 180 days or longer creates a (rebuttable) presumption of discontinued residence, and a single absence of a year or longer creates a legal determination of discontinued residence. Even if no single absence is as long as 180 days, if a Permanent Resident is outside of the U.S. more days than he/she is inside the U.S., that person can be regarded as having abandoned residence in the U.S. Based on this, a single absence of 2 - 3 months will not create any problems about continued residence or physical presence for naturalization eligibility. Some immigration law firms, including mine, offer legal services on a "flat fee" basis so that a client will know the total expense from the very beginning, and a few immigration law firms, including mine, offer an initial consultation free of charge.