For example, paths, roads, bridges used by ancient empires such as the Greeks, Romans, or Chinese dynasties that are still used today? Like an ancient roman road used to lie but now there is a major highway or railroad? Or bridges that were maintained over the centuries? Was there continued development on these roads even after their empires fell?
Sure. The King's Highway in Jordan immediately jumps to mind.
Was there continued development on these roads even after their empires fell?
I'm less familiar with whether or not the development would have been "continuous" per se (i.e. whether or not the road ever fell into prolonged periods of neglect or disuse. I strongly suspect it would have.) but it has been used and developed by a succession of empires that controlled the region, down to the present Hashemite Kingdom of Jordan.
There are also roads that by virtue of simple geography have remained in use in the same place, like the Khyber Pass in the Hindu Kush between Afghanistan and Pakistan.
Roman Emperor Trajan put an inscription on a bridge in AD 98: Pontem perpetui mansurum in saecula (I have built a bridge which will last forever). (Credit to Wikipedia for the inscription and translation). The question is the same as Theseus's Ship--at what point does a bridge cease to be the same bridge? This bridge in Alcantara, Spain has been broken and blown up on a number of occasions in the past couple thousand years...but do the repairs make it the same bridge--or is it a different bridge now?
You decide. Regardless, I think that it is pretty neat that there's a bridge that was commissioned by a Roman Emperor out there.