If you have unintentionally offended someone by finding a mistake, the best course of action is to apologize sincerely and try to understand their perspective. Be open to discussing the issue and finding a resolution.
People may feel offended if they perceive your correction as criticism or as a personal attack. They may also feel embarrassed or defensive about the mistake being pointed out.
When pointing out a mistake, be mindful of your tone and language. Use neutral and non-judgmental language, and avoid using words that may come across as condescending or belittling. Also, offer your correction in a private and respectful manner.
It's important to respect the other person's feelings and boundaries. If they are not ready to accept your apology, give them some space and time. Continuously apologizing or trying to force them to accept your apology may only make the situation worse.
Reflect on the situation and try to understand what may have caused the offense. Make a conscious effort to communicate with empathy and tact in the future. It's also important to listen and be open to feedback from others about how you communicate and interact with them.