to not give up on a relationship when it is at its worst
I don't understand this, not in the context of an affair. It's invalid - the point is that the 'relationship' you have - or had - was one based on trust, love, it was a relationship with someone you were sure would put you first and not be unfaithful. That IS the relationship. When someone cheats, and you walk away, you're not 'giving up on the relationship'. You're recognising that the relationship you had, the one you nurtured, is - for you - no longer there. They smashed it to bits.
This is I presume why those who do decide 'not to give up' often end up in such an empty place. The core of the relationship is no longer there and no matter how hard you 'don't give up', you cannot bring it back. You can't undo the fact that the other person in that relationship destroyed it, demonstrated in the most devastating way they could that it didn't matter to them.
Same for 'learning to trust again'/'getting the trust back'. WTF? The point of trusting someone is, um, knowing that they wouldn't do the bad thing. So - they do the bad thing. Then apparently it's a good idea to work verrry hard to make yourself believe again that they are the kind of person who wouldn't do the bad thing 
You can forgive. You can build a new relationship. But it won't be the same relationship you did have, it WILL be a less special, harder and meaner relationship and you won't ever trust them again - it's nuts to say you would.
It can be worth doing in many circumstances, I am sure, and I can see why people do. But - see it for what it is, or you'll send yourself mad.