Donald Trump is not yet President of the United States.
In fact technically, he is not yet even President-Elect of the United States.
The process of electing a President only begins with the votes being counted on the night of the Tuesday after the first Monday in November. Those are unofficial totals. They don't become official until the Secretary of State or equivalent official in each state and D.C., where the 51 separate elections for President are conducted, certifies them.
Once the totals are official, each state's slate of Electors is notified, according to the laws of their respective states, which will be called to cast a vote for President. Maine and Nebraska apportion Electors both statewide and by congressional district; all other states and D.C. go winner-take-all. Since the Constitution empowers state legislatures to decide how Electors are to be chosen, either option is legitimate -- as would be letting the sitting legislature at the time name Electors without consulting voters at all.
Once the 51 slates of Electors have been named, they are called to their respective state capitals (or appropriate venue in D.C.) to cast their ballots next Monday, December 19.
Once cast, the votes are then sent to Washington to be counted before a joint session of Congress, which is set to occur on January 6, 2017.
For all but two weeks before the Inaugural Ceremony on January 20, there is no "president-elect." There is a presumed president-elect, and given the stability and predictability of the process over the centuries we have grown accustomed to acting as though the presumed president-elect is, in fact, the president-elect. So far it has worked out that way.
Only after the President of the Senate -- in this case Vice President Joseph R. Biden -- announces the result of the Electors' votes will Trump (barring some surprise) actually become President-Elect. As a result, at 12:00 noon Eastern Standard Time on January 20, 2017, he will become President of the United States.
Under the Constitution as amended, the term of the incumbent President ends at precisely that instant and he loses the powers of the office immediately; there is no circumstance under which, constitutionally, he can retain them. His successor cannot take up those powers until he has sworn the Inaugural Oath, but he already holds the office.
If the shit hits the fan and there is no President-Elect by noon on January 20, there are provisions in place dictating who shall exercise the powers of President after that time, and what must be done, if anything, to finalize the succession. Again, the outgoing incumbent can't continue in office if he is not the President-Elect.
One of the more contentious elections of recent times took place in 1960, in which Senator John F. Kennedy, Democrat, ultimately defeated his Republican opponent, Vice President Richard M. Nixon. Despite mutterings about election fraud favoring Kennedy, Nixon solemnly carried out one of his last constitutional duties on January 6, 1961, announcing his former election opponent as President-Elect of the United States.