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Eveline is the main character in the story of the same name. In the story, she is about to leave her family and Dublin to marry a young sailor, with whom she will go to Buenos Aires. She struggles with whether she should stay in Ireland or not. Before her mother passed away, she had promised her that she would take care of her father. She is also very insecure about leaving the place that she knows best, even though her life is pretty miserable there. This is the famous "paralysis" that goes into Joyce's stories.
In the end, Eveline does not go with her "fiance." She holds onto the railing at the dock, "clutching" it "with frenzy." The last paragraphy tells of both the paralysis and the epiphany:
He rushed beyond the barrier and called to her to follow... She set her white face to him, passive, like a helpless animal. Her eyes gave him no sign of love or farewell or recognition.
Eveline has decided not to go. Her epiphany is that her life will remain the same. You may also note that the sailor didn't keep himself off the ship. He shouts from aboard it. That begs the question: Did he really want Eveline to go with him?