Is it true that if a holomorphic function in the unit disk converges uniformly to the 0 function some connected arc of the unit circle, this function is globally null?

If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)

If that is true, this would stand for a uniqueness fact and so how to recover some holomorphic function from its boundary value on an arc (if the uniform convergence still holds on the arc)