As was already said the fuel pump only runs for a few seconds while cranking or when the ignition is turned to run (depends on the car).
Once the rpms reach a certain point it kicks on full time.
Always check for spark too. No spark & no fuel is probably not the pump.
Check for voltage while someone else tries to crank it. If you get voltage for a few seconds but the pump doesn't start it is the pump.
You could also check for voltage at the relay while cranking.
I have also hot wired the pump through the relay connection to test the pump. Also a great way to empty a gas tank filled with diesel by a friend's daughter too.