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I'm not sure I would call the battle unwinable at all. Mrs. Dubose did win her battle to be free of morphine. However, she knew that winning the battle would mean nothing in the big picture. She was dying and getting free of the morphine would not change that. The win would prove something about her will, and about her ability to control her own destiny.
In Atticus' case, he knows he will lose the original court case. He has no doubt of that. However, he still takes the case and hopes to win it on appeal. Like Mrs. Dubose, he also knows that winning Tom's case on appeal will not win the war against racism. Had Tom not been killed, Atticus likely would have gotten him off on rape charges, but the man would have had to leave town or he simply would have been lynched.
So, both people have courage to face an overwhelming obstacle, and both know that winning the battle will not win the war. In that, they both have courage.
In the novel To Kill a Mockingbird Atticus Finch calls Mrs. Dubose, "the bravest woman he has ever met." It is because of Atticus' understanding of Mrs. Dubose's addiction to morphine, and her subsequent battle to free herself from her addiction, that he is able to transform her from an embittered, racist crone to a courageous hero. By taking on the Tom Robinson case Atticus, like Mrs. Dubose, takes on an unwinnable challenge. Both characters are aware of their own, and societies, limitations and yet they both proceed to fight battles they know they cannot win (at least without paying a heavy price). Atticus however, champions his cause not for himself, but rather for his children. It is his own moral code that pushes him to risk everything for the sake of his conscience and his desire to raise Scout and Jem with honesty and integrity.