I don't always use a whole onion when cooking/making a sandwich/etc. And I hated the way that after cutting, the exposed inside part would get dried out or bad.
So I started cutting my onions this way, and would be able to save them for MUCH longer, only using little bits at a time.
I drew this little guide to show how it's done. I hope you find it useful, and don't have to throw away as much old onion in the future ;)