Try wrapping it in foil and store on the top door shelf of you fridge, ( for up to five days). When you come to use it again thinly slice off the dried edge and discard. The fruit will still be fresh and juicy inside.
You can actually now buy little containers that are shaped like "half" an onion, or lime etc. They are plastic and keep the whole thing airtight! You can find them in those kitchen "innovation" type catalogue that so wonderfully come falling out of anything I read nowadays........grumble grumble........