To give up or abandon hope, plans, or an effort; to resign oneself to something.
試験に落ちても諦めずに次に向けて勉強を続けたほうがいい。
Even if you fail the exam, it's better not to give up and to keep studying for the next one.
彼は夢を諦めて会社勤めを始めた。
He gave up his dream and started working for a company.
病気の治療法がないと聞いて、家族は一時は諦めかけた。
When they heard there was no treatment, the family almost gave up for a while.