In the sentence:
what is the function/meaning of きれる? I assume the sentence means "Tanaka didn't give up", but how does it differ from 諦めなかった？
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The verb 切る, besides meaning "to cut", can also be used as a suffix to a verb in the pre-masu form (i.e. 食べ) to mean "to do something completely".
Therefore 諦めきれない would mean "cannot completely give up".
Other places where this is used are verbs like 待ちきれない (can't wait) or 理解しきれない (can't completely understand).
Here is a page which has a lot more examples of ~切る (written by me)
EDIT: Kentaro Tomono pointed out highly that I had not responded to the original poster's question about the difference between "諦めなかった".
As 諦めなかった simply means "he didn't give up", it lacks the meaning of "not being able to" which is present in 諦めきれなかった.
A more interesting question would be the difference between 諦められなかった. Here, I would say 諦めきれなかった, having the concept of not "completely" able to do something, would have greater emphasis. Or it might be more appropriate to say he was torn between giving up and continuing on, so at some point he had partially given up (諦めかけた）but never completely.
I think changing the original sentence to simply use 諦められなかった would be OK and not have a major change in meaning.
Ahm.... I think I would like to say, it is Yes or actually in the end No.
From the link
2 meanings of 「切れる」
About the verb,「切れる」，there are 2 -- derivative -- meanings. The original is "to cut" and it turned into the potential verb, one is transitive, the other is intransitive.
( Your question's is intransitive, considering the nature of the meaning of the sentence )
So here, the verb 「切れる」 is used as the potential verb, and the meaning of the sentence is
"Tanaka san could not give up".