My Hyundai Elantra won't start

My car has had trouble starting since one day when the
weather was -40. I had a mechanic come a jump my car
and he said everything looked fine, my alternator was fine.
The dash in my car will light up and it will beep when I put
the keys in the ignition but it won't start. I replaced the
battery thinking I just had a dead battery. My car was
working fine for a couple weeks and started with no
problem but now I'm back to having to give it a boost
everytime I want to use it.