I don't know of any car made that will not run on the alternator even if the battery is dead (but still connected of course). On the older cars you could remove the negative battery cable and if the car died then your alternator was bad but in the last 20 years or so the computer will shut the car down even if it is good.

However if your battery is dead and you jump start it what do you do? You let it run for a little while to charge the battery right? Therefore the car is running off the alternator while it is also charging the battery for ya.

If your car will not remain running then I would almost bet money that your alternator is bad. One way to tell is to start your car and turn your headlights on bright, A/C, stereo, wipers and every other accessory that you can think of. If it dies then you will know that your charging system is not putting out enough. You can try Auto Zone or Advance but I have had several test good that were bad and vice versa.

If you have a volt mete stick it on the battery post (one on positive and one on negative of course) and see what the voltage is while the car is running. It should be over 14 volts or maybe more depending on the make with the lights on.

Good luck and hope this helps.