give someone up
1. To yield or relinquish something. A noun of pronoun can be used between "give" and "up." This is a prime parking space—I'm not giving it up! It would be a shame if you had to give up this apartment—it's such a great location.
2. To abandon some task or goal. Oh, don't give up—you'll get this boiler fixed eventually.
3. To stop doing something in particular, often a habit. Lots of people have given up smoking, so I'm confident that you can too.
4. To surrender or yield someone or oneself. In this usage, a noun or pronoun is used between "give" and "up." You need to give yourself up to the cops before an innocent person gets caught in the crossfire. I know he's done some really heinous things, but he's still my brother—I can't just give him up to the authorities!
5. To lose faith in someone or something; to no longer believe in someone or something's potential for betterment. Always followed by "on." I know I've failed a lot in the past year, but please, don't give up on me. Don't give up on this team just yet.
6. To fully pledge or devote oneself to something. In this usage, a reflexive pronoun is used between "give" and "up," and the phrase is always followed by "to." To deal with the grief, I gave myself up to my research.
7. To consider lost or without hope. A noun or pronoun is used after "give" or after "up." Nearly always followed by "as lost" or "for lost." I gave the pizza up for lost when it didn't arrive in two hours. We had nearly given up the situation as lost when help arrived.
give someone up
tv. to betray someone; to turn someone in to the authorities. No, I didn’t give Mooshoo up!