You have a flat battery (as you already know) but I'm suspecting that there is probably either low or no electrical power coming from your alternator. Even with a flat battery, once started the car should keep running because the alternator is supplying the electrical power.
In your case, check that the fan belt is not broken or loose. If it is replace it or tighten it, jump start your car and see if it keeps running, with the jump cables removed. If it runs, take it for a drive to put a bit of charge back into your battery, so that it will start next time
If the fan belt looks OK do the following.
If you have a voltmeter, connect the other car's battery with cables to your battery, do not have the other cars's engine running, start your car and measure the voltage across the your car's battery terminals, with the jump cables still connected. If you only read about 12V (you are reading the other car's battery voltage- not your alternator) the alternator is faulty (it is probably the regulator but that is inside the alternator anyway). It should read 13.8-14.5V approx.
If you have no voltmeter you need to get an auto electrician to check it out. If you have one close by, if you replace your battery with a charged battery it should get you there before it goes flat - maybe ( make sure that you have the lights, radios any thing electrical turned off, - to take the drain off the battery, it needs to power the car), or if there is a mobile auto electrician who comes to you or get it towed there.