This is the 11225th most frequent Russian word.
давшего
"Давшего" - "of someone/something that gave" (past tense, genitive case, singular).
"Давшего" is used as a past participle describing the engineer in a masculine form.
Город благодарен инженеру, давшего новой технологии вторую жизнь.
The city is grateful to the engineer who gave the new technology a second life.
"Давшего" is employed to specify the man in a genitive clause.
Мужчина, давшего будущему поколению надежду, останется в памяти людей.
The man who gave hope to future generations will remain in people's memory.
"Давшего" modifies "писатель" and is in the genitive case consistent with the syntactic rules.