I've been wondering about the tenacious badge for some time, now I've been informed I got it. I understand the condition of winning the badge, but what is its purpose? What does it reward?
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It's kind of a consolation badge. It's recognition that you post useful, working, accepted answers even though at least 5 of these answers went without the upvotes they may have deserved.
An answer accepted with a score of 0 sounds to me like a working answer that didn't get the upvotes it was due. That is what this badge rewards.
Everything John says is correct, but in addition: