Translation guide
To stop believing that someone or something will improve or succeed, or to stop trying to help or change them.
To lose hope or faith that a person will change, improve, or succeed.
To give up on someone after losing hope or trust; implies a final decision.
He lied so many times that I finally gave up on him.
To abandon or forsake someone, often emotionally; stronger than 見限る.
a child given up on by their parents
General 'give up' but can be used for people when you stop expecting change.
I've given up on him.
To cease efforts to assist, reform, or influence a person.
To withdraw support or involvement; literally 'pull one's hand back'.
I've given up on that student.
Idiomatic: 'throw in the spoon' (like 'throw in the towel'); give up on a hopeless case.
an incurable disease that even doctors give up on
To abandon or desert someone in need; stronger nuance of leaving behind.
To lose hope that a project, idea, or object will work out.
見限る implies a rational decision after disappointment; 見放す is more emotional abandonment; 見捨てる is physically or morally leaving someone in a bad situation.
Avoid directly translating 'give up on' as あげる＋上に. It doesn't make sense in Japanese.
困っている友達を見捨てるわけにはいかない。
I can't just give up on a friend in trouble.