Today is the time to talk about giving up.
The second of the questions is easier to answer. For tradition has it that believers should imitate the self-sacrifice of Christ who spent forty days denying himself in the wilderness in order that he might become Savior of the world. So as early as the fourth century the practice of fasting began, as a sign of piety. Yet there is a fundamental element in the story of Jesus' temptation in wilderness which seems just the opposite of giving up. It consists of the fact that he didn't give into, or succumb to the enticements of Satan.