Abiding seems to be a crucial part of Katergazomai.
As we "work out our faith" we have to abide.
Abiding means living like his Sons and daughters. It means seeking out fellowship with Him. When we love God, we want to be close to Him. We want to live in His ways. But it does take work, it does take intentionality. It does take moments when we chose to sit down with a Bible study, when we chose to change the chanel or listen to a different type of music, when we chose to go to church, to fellowship with other believers,to learn more about Him. To me, abiding in Christ is katergazomai. It is working out my faith. Works don't make us a Christian, if we could work hard enough, we would, but we needed an ultimate sacrifice, Jesus. Just the same, works reveal our faith. Reveal our true heart. Yes, those works may be imperfect, but I would hope that, over time, the way we act, the way we speak, would only become more like Jesus.
You know the Jars of Clay song, "They will know we are Christians by our love?" (also see John 13:35), that's the song running through my brain at the moment, but I think that's it at the core. We shouldn't have to scream it, we should be able to display it. Imperfectly admit to our faults, but also express our true heart intentions.