Ever since I was a kid, I've been attending summer theatre plays in Lyon's Ancient Theater, which is currently 2029 years old and still used for its intended purpose. There must be even older buildings or other features (aqueducts, maybe?) still used today in Italy, Greece or perhaps Iran or Iraq, so I was wondering if anyone had attempted to determine what the oldest such site would be.
Waiting for a professional historian here you may argue that structures like Stonehenge and Egypt pyramids still serve its original purpose.
We do not really know what Stonehenge purpose was, if it was let's say "a monument to the Sun" then it would still be serving its original purpose, same may be argued about other Dolmen like structures.
Egyptian pyramids are still tombs, and as far as I know remains may potentially be there since they've not been explored entirely and additional undiscovered passages and chambers may exits.
As I say there is a bit of guessing here but if we do not know the original purpose of something then it may still be there playing that role. I look forward to see other responses from more qualified people.