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In the video announcing his presidential run, Bill de Blasio, the mayor of New York City, described his hometown as “legendarily tough and big and complicated.”

Soon after, New Yorkers proved just how apt that was.

As Mr. de Blasio officially kicked off his campaign for the Democratic presidential nomination on Thursday, the reactions of city residents ran the gamut from support to indifference to vehement objection.

“Bad idea, Mr. Mayor,” Sharon Henderson, 53, said.

“How do you want to run off and be president of the United States when you’ve got all these issues you’re supposed to be the mayor of?” said Ms. Henderson, a home health aide who works in Brooklyn and lives in a homeless shelter in Queens. “Nah.”

Mr. de Blasio, the 23rd entrant in the Democratic race, is facing the challenge of standing out in a packed field of candidates, many of whom already have significant leads in polls and fund-raising.