Does the Cat must be replaced?

My old Civic just got front tires replaced two months ago and battery replaced one month ago - now the exhaust pipe right after the cat was completely broken and cam off due to rust -- tried to 電焊 together once but could not last for a month. Contacted the owner of the repairing shop where my old Civic had tire replaced before, I was told that the Cat may have to be replaced so that a new pipe can be  電焊 together with the cat. The question is, the cat is still fuctioning well but got rusted bad at both ends, is it a must to be replaced, or is it a must in the exhaust system for the vehicle to run normally? P.S: the state I live now does not require annual inspection.  Just wan to keep the old car for maybe two to three more years as no budget for another new car yet.

Thanks for any suggestion/advice.