As many have noted, whitehouse.gov was changed over to reflect the new administration before the oath of office was even taken. I’ve got no problem with that; Obama became president at the stroke of noon Eastern, regardless of the sequence of the ceremony.
I am just a wee bit dubious about this, however:

First, the campaign is over. He’s now president of the country. Why does it look like his campaign website just changed URLs?
Second, not to be nitpicky, but there was no such person as “President Obama” on January 19.
Looking at the Wayback Machine to confirm that George W. Bush didn’t convert the official website of the presidency into a campaign website, I was reminded of how rudimentary sites were back on February 2, 2001, the first archived day of the Bush presidency:
And, no, it’s not just Republican technological inadequacy. Here’s what it looked like on January 18, 2001:
Even at 600 pixels, the site’s blurry because it’s nothing but text and links. Small text with undifferentiated fonts, unless one counts boldface.











