Atticus Finch displays courage before the lynch mob, who pull up near the jailhouse, by appearing as though he is relaxed, sitting in his office chair propped against the building, reading his newspaper. Then when the Old Sarum Bunch pulls in, he acts as though he were expecting them. 

In Chapter 15, after the principal figures of the town visit Atticus the night before to ask him to procure a change of venue for the Tom Robinson trial, Atticus suspects that there may be foul play. So, he goes to the jailhouse and sits before the cell holding Robinson. When the four dusty cars of the Old Sarum Bunch, reeking of whiskey, pull up before the jail, Atticus sits calmly and looks up from his newspaper. Very deliberately he folds it and drops the paper into his lap, pushing back his hat at the same time.

When the men ask if Tom is inside, Atticus calmly answers, "He is...and he's asleep. Don't wake him up." Ironically, the men actually speak in whispers.

"You know what we want...Get aside from the door, Mr. Finch."
"You can turn around and go home again, Walter....Huck Tate's around somewhere."

Mr. Cunningham contradicts Atticus, saying that the sheriff is "out in the woods" because the men have tricked him. But, Atticus remains calm, saying that changes things. "It do" a shadowed voice affirms. With his classic question, Atticus pointedly asks, "Do you really think so?"

It is then that Scout and Jem intervene. Frightened for his children, Atticus folds the paper again, but his hands tremble. Still he remains calm.

Fortunately, Scout diffuses the situation when she speaks to Mr. Cunningham, and he is ashamed for having threatened Atticus Finch who is showing courage by his calm and by his strength in facing the mob down. Cunningham respects Aticus's courage and tells the others in the mob to leave with him.

How does Atticus Finch display courage in the face of persecution?

Atticus displays immense courage by agreeing to represent Tom Robinson in his forthcoming trial for the rape and assault of Mayella Ewell. He knows it's an unwinnable case; Tom is a black man accused of raping a white woman, and at this time and in this place, that's enough to convict him in the eyes of Southern society before he's even set foot in court. Atticus knows that he and his family are going to be given a hard time by other folks in Maycomb because of his decision to act as Tom's defense attorney. But Atticus also knows that defending Tom is the right thing to do, and so he bravely goes ahead and gives Tom the best legal representation that he can possibly give.

Once he's made his decision, Atticus sticks by it, even when it endangers his life. Atticus comes close to getting himself seriously injured or maybe even killed when an angry, drunken mob led by Walter Cunningham turns up at the jailhouse to lynch Tom Robinson. When they arrive, Atticus is sitting out front calmly reading a newspaper. But the atmosphere soon turns ugly, and although he must be absolutely terrified at the sight of all these fearsome-looking men with hate in their hearts, Atticus doesn't back down and refuses to step aside to let the lynching party gain access to the jailhouse. He's absolutely determined that Tom will have his day in court, even if it means putting himself in harm's way.