Definition: to stop doing something or believing something. (This phrasal verb has more than one meaning)
E.g.1: She wanted a new dress and asked her father to buy it for her. Every time she asked him he said no, but she continued to ask him every day. Eventually she gave up asking him and never got the dress.
E.g.2: My doctor tried to persuade me to give smoking up.
This phrasal verb can be separated. (E.g. 2)