For a car with fuel injection stalling during idling is either caused by a bad idle air bypass valve, bad cold start valve, bad fuel pressure regulator.
Possible reasons why the car is stalling at idle
(1) Bad idle air bypass valve.
(2) Bad cold start valve.
(3) Bad fuel pressure regulator.
Car troubleshooting
If the engine stays running if you lightly step on the gas pedal, then idle air bypass is faulty.
As it is indicated that the car has fuel injection. Fuel-injected engines have a device called an "idle air bypass valve". This valve is responsible for maintaining the engine's idle speed. Hence, if it's not working, the engine will stall.
Now then, since stepping on the gas pedal will keep the engine running when it's about to stall. This clearly indicates a faulty idle air bypass valve. You see, by stepping on the gas pedal just as the engine is about to stall, you are essentially compensating for a faulty idle air bypass valve in that you're increasing the throttle and thus giving the engine enough fuel to stay running.
What to do?
You'll need to have a mechanic verify that the idle air bypass valve is faulty. If this is indeed the case, have it replaced. In a situation like this, some mechanics may offer to replace several other items related to the above. For now, just stick with the replacement of the idle air bypass valve. This should correct the problem. You can always return on another date for the other items the mechanic suggested. We recommend AAA approved garages since they tend to be very reputable.
However, if the engine did not stay running if you lightly step on the gas pedal then rule out the faulty air bypass valve as the culprit. Next is to observe if the stalling occurs when the engine is cold, if so then it is a problem with cold start valve.
As you indicated above the car is equipped with fuel injection. Fuel-injected engines have something called a "cold start valve". This valve keeps the engine running whenever it (the engine) is cold. Basically, it achieves this by giving the engine extra gas for a richer air/fuel mixture. Now, as you've indicated, this stalling problem only occurs when the engine is cold. Clearly then, this points to a problem with the cold start valve.
What to do?
You'll need to have a mechanic verify that the cold start valve is faulty. If this is indeed the case, have it replaced. In a situation like this, some mechanics may offer to replace several other items related to the above. For now, just stick with the replacement of the cold start valve.
This should correct the problem. You can always return at another time for the other items the mechanic suggested. We recommend AAA approved garages since they tend to be very reputable.