It doesn't exactly matter now, as Obama took the oath again yesterday just to be sure and silence those critics who claimed he wasn't President because of the bungled oath on Tuesday. But, just as a point of fact, those people are wrong. Obama was President at noon on January 20th regardless of whether he took the oath or not. What he could not, technically do, was exercise the authority and powers of the Presidency without the oath.
The Constitution, in Article II, states: "Before he enter on the Execution of his Office, he shall take the following Oath or Affirmation: 'I do solemnly swear (or affirm) that I will faithfully execute the Office of President of the United States, and will to the best of my Ability, preserve, protect and defend the Constitution of the United States.'"
Note what it says: Before he enter on the Execution of his Office...
That's a bit different than saying, "Before he becomes President..." Some have argued that the Office and its powers are the same, and that the Founders therefore meant, essentially, "Before he becomes President...", in disregard for the actual text.
But, supposing we humor them, it still doesn't matter. Because the Twentieth Amendment, ratified in 1933, provides that: "The terms of the President and Vice President shall end at noon on the 20th day of January... and the terms of their successors shall then begin." Even if the original text of Article II really meant that you couldn't become President without taking the oath, even though that's not what it says, in light of the Twentieth Amendment it is clear that Obama became President on noon of January 20th. The Twentieth Amendment doesn't mention anything about an oath.
However, technically, I do believe that Obama didn't actually have authority to exercise the powers of the Presidency until he took the Oath, subject to Article II. So, while it is untrue that there was ever a gap in Presidencies, there was, technically, a gap during which no one had authority to exercise the powers of the Presidency. Even though there was a legal Vice President, Joe Biden, and failing to take the Oath could certainly count as a "disability" that would allow Obama to be removed from office if he refused to re-do it, there's still a process Biden would have had to go through before he could assume the duties of the President, including, of course, taking the Oath himself.
Interesting little tidbit of constitutional law there, but I think, reading the text, you'd have to really want to read something into it to get out of what is there that Obama ever wasn't actually President after noon on Tuesday.