The sentence "my father didn't give up on me" means that your father didn't stop believing in you. The phrase "give up on (someone)" means stop believing in or having faith in someone. The sentence "my father didn't give me up" means that your father didn't give you away, as in, give you up for adoption or give you away to law enforcement (police, or others), etc.

well, the another one... give up... is a phrasal verb, which means to stop trying, or doing something. (surrender, abandon habit, addiction, etc.) the order of the particle it doesn't matter. Examples: I want to give up smoking Many women give themselves up to raising their family. ( Muchas mujeres se dedican enteramente a criar a su familia.) My father didn't give me up.- Mi padre no se daría por vencido (easily with me, because is my father, he loves me, and he's trying and trying with me) Well. I hope I've been able to help you. as I told you. I'm a student too. Maybe a native wants to give us a wide explanation, to clear your doubts if I've made mistakes.

What's your base language or mother language? your profile say English, and you're learning English too... I'm asking you this, because, to give you an answer I don't know in what language give you a reference. Mine is Spanish.

I'm an English student too, my contribution is... give up on... is an idiom, it refers to "abandon someone"... check this lines: And the first time I ever felt like that was with you. Y la primera vez que sentí eso en toda mi vida fue contigo. And I don't want you to give up on me. Y no quiero que te rindas conmigo. I won't, Virgil. No lo haré, Virgilio. , When I was in Atlanta I realized how much that I need you. Cuando estuve en Atlanta, me dí cuenta de lo mucho que te necesito.