GFCI that opens when power source cut
This is found on plug-in GFCIs

The behavior you are describing is found on GFCIs that are intended to be plugged into a receptacle, like the one depicted below, instead of hardwired into building wiring.  It is necessary to prevent the combination of an open neutral in the building wiring and a ground fault in the appliance from causing a shock hazard the GFCI cannot cut off (because it has no power to its electronics in that case).