Granted, it is important for a child to know their father. It doesn't matter the circumstances under which the child was born. Doesn't matter even whether the two parents are in talking terms. It is fundamental that a child grows up knowing both the parents if they are alive. And in the event that either of the parents is not alive, then the surviving parent has an obligation to bring to light the circumstances under which the child was born.
There is a quest in each one of us to know our origin. Children that grow up not knowing one of their parents are not complete. No matter how well they look and make it in life. And it is more paramount when it is the dad that is unknown. Ultimately, children belong to the father. Even the Bible says so. It is the father that gives life to the child. Without him there would be no formation. Not discounting the importance of the mother, but that is the truth.
And so it is futile for a mother to withhold the identity of her child just because they no longer relate with him. She is actually doing her child disservice. The better way would be to sit her child down once he is old enough to understand, explain it all to him and if the child expresses a desire to know the dad then she finds a way for the two of them to meet. Despite how she feels about him or how much she has moved on with her life. It is for her child's sake. If he does want it. And he will. Maybe not then but some day the urge does come. Blood calling out?
I have seen some kids brought up by single mothers who do a real good job. They take them to real good schools, work hard to ensure that the child is well provided for. But the kids grow up with a vacuum. They seek to fill it in various ways. Some not so appealing. Of course some kids grow up entirely not wanting to find out who their dad is. And live good lives of their own.
The excuses fathers give for not knowing their kids are varied. And comical sometimes. It is pure lack of initiative on their part that they do not know of their offspring. No matter how many miles apart. Any father who genuinely wants a relationship with their child will find  a way. Could be through the courts or warming the baby mama's heart. Once again. This will cushion the child of growing up with a void of if only and him of guilt trips. And so it makes it easy to deal with any eventualities that may come up in life. This way, the baby mama does not have to have to deal with the trauma of God-knows-how-many dads coming up to lay claim of the child when he is all grown up, has made a name for himself (and wealth too). And life is much easier for all.
I once witnessed a scene that broke my heart. Of a little girl who was en broiled in a split up of her parents. Even as she clung to her mom she was reaching out to daddy, all cries. But he was all packed and leaving. The heat was too much for him to take. And whereas he was moving on to a new life, he was leaving behind a broken heart scarred for life. It was all over her face. She needed her daddy.
In the face of broken relationships it might help to stop and reflect on the impact that will have on the kids if any. And if it is inevitable then ways of helping the child cope with the issue should be sought. Mainly this should not be difficult except that the two parents usually put their own feelings at the fore. Thus overlook the vulnerable kids that are left torn and changed for life. An arrangement can be reached when the two never lived together in the first place. With the well being of the child top most, an amicable truce can be reached allowing the child to be with the two abeit in separate circumstances.