Here’s the honest truth: a cut onion is best used within 3 to 5 days. After that, it starts losing its crunch and its flavor gets all mopey. If you’ve stored it correctly, it will still be safe to eat for up to a week, but the texture won’t be as nice for eating raw on a sandwich.
If you see any black fuzz or smell something that’s not “onion-y” but more “ew,” toss it. Your nose knows. But if it’s just a little soft? Chop it up small and cook it into a sauce or a stir-fry—cooking revives even sorry-looking onions.