There's some arguments on the subject, most of which can be found scattered through here. Zachary Taylor declined to be sworn in on the day Polk's term expired (it was a Sunday), so there's a standing legal question as to whether he was nonetheless president that day, whether someone else temporarily held the job, or whether it was vacant. (Since it WAS a Sunday, the executive branch wasn't doing anything anyway, so no one needed a president and no one had to figure it out for reasons other than mental exercise.)