05 century starts but does not catch

My 05 century has been randomly not catching when i start. The engine immediately cuts out. If I give it gas it remains on and sounds terribly rough. After I let it run by holding my foot on the gas it slowly becomes less rough and then the engine remains on and goes back to its usual smooth self and starts and catches. Then it wont do this again for awhile. Recently it has started doing it more often. I can drive it by holding my foot on the gas. but not very safe.
It sounds like bad gas but I have gone through a couple of tanks and different gas stations.
I had it checked but my mechanic could not find any codes. He checked the fuel lines and cleaned the the air and throttle body and cleaned the maf.
I am at a loss of ideas. Any help or suggestions would be greatly appreciated.
I forgot to mention it never cuts out if I give it even a drop of throttle
I recently had a new fuel pressure regulator put in. Is it possible that that is the problem.

What is the reading on a fuel pressure gage(turn key on, then off....purge air....then turn on, shut off for 10 seconds, then repeat 3 times to get max pressure)? Does it hold, or does it drop quickly, or over a 5 minute time period? This has to be checked when the problem is happening....

By "on" you mean on or start.
The mechanic checked the fuel pressure and said it was fine although it was starting and catching then fine because I had just driven to the shop with my foot on the gas and then it started working again.
After yesterdays ordeal with the issue I actually got a code - lean fuel mixture-

It only has the problem sometimes and after I drive it with a foot on gas and brake to the mechanic it is usually gone. Once I drive the roughness clears up. My mechanic refuses to change the fbr without seeing the problem. He checked the pressure when the problem was not happening and it was fine. Any suggestions on a test that would show the problem?

The problem is that it doesn't happen for your mechanic.....could be something crazy like water in the gas, that gets through intermittantly...really need a scan tool on this when it happens.....

By the way, when I say on, it means key on, engine off(no cranking)....

try disconnecting the MAF and then start the car. If it runs better with the MAF unplugged, it is your issue. On these cars, when the MAF first starts to fail, it will exhibit your exact symptoms, but won't trip a CEL. As it gets worse, it usually will trip a MAF input voltage low code first, but can also trip any one of many MAF related codes.

If unplugging the MAF doesn't change how it runs, you need to check for vacuum leaks. Common vacuum leak areas on these cars is the rubber boot where the hard plastic lines attache near the throttle body, the vacuum line that runs near the battery tray area for the HVAC system, and the one for the purge solenoid.

try disconnecting the MAF and then start the car. If it runs better with the MAF unplugged, it is your issue. On these cars, when the MAF first starts to fail, it will exhibit your exact symptoms, but won't trip a CEL. As it gets worse, it usually will trip a MAF input voltage low code first, but can also trip any one of many MAF related codes.

If unplugging the MAF doesn't change how it runs, you need to check for vacuum leaks. Common vacuum leak areas on these cars is the rubber boot where the hard plastic lines attache near the throttle body, the vacuum line that runs near the battery tray area for the HVAC system, and the one for the purge solenoid.

What makes me think it is not the MAF is the fact that when I keep my foot on the gas and keep it from cutting out it slowly over 10 -15 minutes smooths out and then starts running fine

I finally got it to my mechanic and he changed the idle air control. He did nit charge me but said if the car is fixed I should come in and pay.
I am an amateur but I can not believe that the symptoms are iac related.