I know it sounds foreign to many but growing up, I've been calling these yummy fruits "pisang kaki" which in direct translation from Malay language to English means "banana foot" (pisang=banana, kaki=foot) but it's not a banana! That's what it is called in Malay. It definitely sounds better in English - sharon fruit or persimmon.
This is another seasonal fruit in Autumn/Winter months. The perk of buying fruits and vegetables during its season is they are much cheaper than buying them at any other times. Have a look at this seasonal chart. Go season!