How we cite our quotes:
Troy: "But I'll say this for him [Troy's father]...he felt a responsibility toward us." (1.4.109)
This is the one good thing Troy has to say about his father. Despite his meanness, he did feel a sense of duty toward his family. It seems Troy inherited this sense of obligation.
Rose: "And upstairs in that room...with the darkness falling in on me...I gave everything I had to try and erase the doubt that you wasn't the finest man in the world...Cause you was my husband." (2.1.122)
Rose talks here about the duty she felt to Troy as his wife. She's done nothing but struggle for years to be the best wife she knows how to be. Now her husband has disappointed her terribly by fathering a child with another woman. In her mind, though Troy has fulfilled a certain part of his duties by providing for his family, he's failed in some other, perhaps even more important, areas.
Troy: "Rose...I'm standing here with my daughter in my arms. She ain't but a wee bittie little old thing....She's my daughter, Rose. My own flesh and blood. I can't deny her no more than I can deny them boys." (2.3.3 -2.3.7)
Troy feels a strong fatherly sense of duty to his newborn daughter Raynell. At this point in the play, he may have to go out into the streets and be homeless with her for all he knows. Still, he's determined to provide for her as best he can. In her typical saintly way, Rose says she will take on the responsibility of raising Raynell. However, she tells Troy that she's not his woman anymore. That's one duty she refuses to fulfill in the future.