The antonym for "give something up" is "hold on to." When we give something up, we relinquish possession or control over something. On the other hand, holding on to something means that we are retaining possession or control over it. When we give something up, we may be acknowledging the inability to sustain or accommodate something or may signify disappointment, as in saying, "I give up!" Conversely, when we hold on to something, it means we are determined to maintain, sustain, or protect it. In general, the antonym of giving something up represents a mindset of persistence, determination, and perhaps even stubbornness.