Jonas shows integrity throughout the book by being honest and demonstrating that he is a good person.
Integrity refers to the fact that someone is a good person. Jonas shows he is a good person not by following rules blindly, but by doing what he feels in his heart is right.
The Receiver of Memory has to be someone who stands out from the crowd. In a community that embraces Sameness, that is quite a lot to ask. When searching for the new Receiver to train, the community elders need to ensure that the person has integrity and can make moral decisions on his or her own, because there are no moral decisions to be made in the community on a daily basis. There are rules governing every kind of behavior. People don’t make choices, so they don’t need to have integrity.
When the Chief Elder is listing the important traits the Receiver has to have, she lists intelligence first and integrity second.
"Integrity" she said next. "Jonas has, like all of us, committed minor transgressions." She smiled at him. "We expect that. We hoped, also, that he would present himself promptly for chastisement, and he has always done so. (Ch. 8, p. 62)
It is hard to determine who is a good person or not deep down, so one way the committee can tell is that when Jonas does something wrong, he owns up to it. This is how they know he has integrity.
Jonas’s integrity becomes an important part of the story when he learns that his community is not as perfect as he thought. The murder of the newborn twin at the hands of his father tells him that something is terribly wrong. He risks his own life and breaks the rules to escape and save baby Gabe, and return the memories to the community so they can make their own choices. That’s real integrity.