I think that there could be a significant debate as to whether or not Troy is entirely irresponsible.  On one hand, he does not show faithfulness or responsibility towards Rose.  He does not show an emotional responsibility towards his children.  He also lacks an affect related to responsibility in his friendship with Bono.  Yet, I do not see him as a character devoid of the trait.  He is a provider for his family.  Even when he leaves to be with Alberta, he does not discard his family, providing for them even when he does not live with them.  His care for Gabriel also represents a form of responsibility.   He also recognizes in the most full way possible his own limitations as a father and husband, as he does not pretend to be something he is not.  Troy is willing to concede that he could be more as a husband and father.  Unlike his own father, Troy actually does try to be a better emotional and psychological provider.  He recognizes Bono's depiction of the "walking blues."  Troy might not be a character that possesses full responsibility, but he does not act devoid of it.  In this, the complexity of his character is revealed.  It is for this reason that Troy demonstrates a complex and intricate relationship with the notion of responsibility.  It is not something that he is entirely clear upon himself, which is why his relationship with it is far from clear, as well.