Suppose a quantum system starts to evolve under a Hamiltonian from some initial state. When, for the first time, will an observable attain a preassigned value? To answer this question, one method often adopted is to make instantaneous measurements periodically and note down the serial number for which the desired result is obtained for the first time. We apply this protocol to an interacting spin system at zero temperature and show analytically that the response of this system shows a nonanalyticity as a function of the parameter of the Hamiltonian and the time interval of measurement. In contrast to quantum phase transitions, this type of phase transition is not a property of the ground state and arises from the Hamiltonian dynamics and quantum-mechanical nature of the measurement. The specific system studied is the transverse Ising chain, and the measurement performed is whether the total transverse magnetic moment (per site) is not equal to 1. The results for some other types of measurement are also discussed. © 2016 American Physical Society.