“To give up” is a phrasal verb which has more than one meaning. 1. To lose hope. 2. To quit a habit or addiction.
Examples for 1:
When starting a business, never give up!
Can you imagen what the world would be like if Steve Jobs had given up after some of his mistakes?
The problem is that most people give up before they even start!
Examples for 2:
After giving up coffee, I actually found I was more productive.
I tried studying Chinese but quickly gave up.
Giving up additions isn’t that difficult if you know how to do it.