should i give up on singing?

Ask your teacher for feedback as to why you got an F.

You shouldn't give up something just because of one setback, even if it's a dramatic one. Nobody would ever achieve anything if that was the case.

Maybe you should ask your teacher why... maybe he/she just give wrong

Yes, give it up! If the first time someone doesn't just pat you on the back and tell you that you're terrific, it tears you apart, then you don't have the mettle to withstand the kind of withering criticism you face in you actually try to make singing your profession. Did the teacher ANNOUNCE everyone's grade, so that you KNOW that you were the ONLY one who failed? If so, that was unnecessary, but if you think you were the only one that got an "F" because you asked the other students, don't expect them to tell you the truth.

Failure is a learning opportunity. Ask for feedback so you know what areas you need to improve. Find out if it was your vocals, or if your PERFORMANCE actually did suck because you either stood there emotionless and sang "pretty" , or you went completely overboard with "making it your own", and destroyed any semblance of what the song sounded like. Maybe you were insufferably smug, and THAT came across when you sang.

But if you have a teacher who only compliments you no matter what you do, then you will learn nothing. It will not prepare you to handle criticism, even UNFAIR or biased criticism.

You might have been rated on YOUR performance. If it was some sort of test or vocal jury, you are rated on your own performance. It's not a contest of sorts where they voters pick the "best singer" of the bunch. Perhaps she looked at how much someone improved over what they could do before. Someone starting at a very low point might get a higher grade if they manage to improve a LOT (which doesn't mean that they are necessarily good). If you sounded and performed EXACTLY as you did before you started lessons, if you went and sang a song your teacher specifically requested you NOT sing, then you get a very deserved "F". None of us here know the circumstances, so the only person who can give you an answer of why you failed your exam, is your teacher.

You need to talk to the teacher. Just giving an F is not enough. A teacher's job is to help students succeed, not just punish them if they fail.

Did you ask the teacher why you received an F? If you're ready to give up because of one set back then perhaps you should give up. That's an indication that you don't have what it takes to make it. Do you know how many times Edison failed before he succeeded with the light bulb or the record player? Did Jennifer Hudson didn't give up because she didn't win American Idol? No, and she went on to win an academy award. So if you don't have any more fight in you than that then yes, give it up.