Definition of give up on (someone or something)
1 : to stop trying to improve the condition of (someone) He has gotten in trouble many times, but his parents have never given up on him. Please don't give up on me. I promise I'll do better.
2 : to stop having hope of seeing (someone) We'd given up on you hours ago!
3 : to stop trying to do or achieve (something) They have given up on their plan to build a new factory. She hasn't given up on trying to convince her husband to buy a new car.
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