Who is Mother/Rosemary from This Boy's Life and what is their importance?

Rosemary is Toby's mother, but she does not get introduced as Rosemary for quite a while. At the beginning of the novel, the author only refers to her as mother. She is introduced as Rosemary when a man who is hitting on her calls her by her first name. The switch from mother to Rosemary in this instance suggests that she has two identities. One she shows for her son and the other she shows to men. Toby constantly competes with other men for her attention and sees her as a different person when she's around men.