Then again, how often does the Unruh-DeWitt detector click if we switch it carefully?

The transition probability in the first-order perturbation theory for an Unruh-DeWitt detector coupled to a massless scalar field in Minkowski space is calculated. It has been shown recently that the conventional i epsilon regularization prescription for the correlation function leads to non-Lorentz invariant results for the transition rate, and a different regularization, involving spatial smearing of the field, has been advocated to replace it. We show that the non-Lorentz invariance arises solely from the assumption of sudden switch-on and switch-off of the detector, and that when the model includes a smooth switching function the results from the conventional regularization are both finite and Lorentz invariant. The sharp switching limit of the model is also discussed, as well as the fall-off properties of the spectrum for large frequencies.