SAN FRANCISCO — The writer Salman Rushdie hit Twitter on Monday morning with a flurry of exasperated posts. Facebook, he wrote, had deactivated his account, demanded proof of identity and then turned him into Ahmed Rushdie, which is how he is identified on his passport. He had never used his first name, Ahmed, he pointed out; the world knows him as Salman.

Would Facebook, he scoffed, have turned J. Edgar Hoover into John Hoover?

“Where are you hiding, Mark?” he demanded of Mark Zuckerberg, Facebook’s chief executive, in one post. “Come out here and give me back my name!”

The Twitterverse took up his cause. Within two hours, Mr. Rushdie gleefully declared victory: “Facebook has buckled! I’m Salman Rushdie again. I feel SO much better. An identity crisis at my age is no fun.”

Mr. Rushdie’s predicament points to one of the trickiest notions about life in the digital age: Are you who you say you are online? Whose business is it — and why?