Breaking up

How do you KNOW when a relationship is over? Like really know. I'm struggling over here. I know for a fact I would have broken up with my significant other a long time ago if we didn't have kids. He cheated but we said we'd try to work things out for the kids. Now a couple of months later he is putting no effort in whatsoever. He's also gone very quickly back to the things he was doing while he was cheating. Making excuses to stay on the sofa for example. I keep telling myself let's give it another month and nothing is changing! He wants to stay together but my heart isn't in it anymore. Everytime I say it's not working he says don't break up the family.