What does giving up mean to you? I believe it depends on what you are talking about.
I am a competitive person when it comes to playing a sport so in that area giving up would never be an option to me though I do believe in team work. Other words in a team sport no matter how we are doing whether winning or losing we must work to accomplish the outcome we want which of course is winning.
If you are in a leadership I believe giving up as in your position is very important. Similar to playing a sport team work comes before anything else. Often teams fall apart especially at the top because the person in charge forgets it is not about him or her but about the team. Giving up your personal ambitions and desires are important. The leader must get the point across to the team that each member must put his personal ways aside for the team. This goes to if you are mentoring someone else as well.
I have found in some cases that the person in charge has a way they have done things in the past and they have a new team still believing every circumstance it will work and doesn’t necessary happen. You have to look at each member of team.
Another area is if you have become a member of a team and you no longer the leader it is important to put your ego aside. In fact this is the most important thing we all must give up is our ego. It is never about us no matter the situation. Not an easy thing to do however success happens only if we can put our self out of the equation. I have known those who have left situations that they were asked to take a lower position because of their ego. Remember that we have something to contribute even when we are no longer in charge. So give up what has been successful because in the future you will have more success.