Originally Posted by Jonathan de Boyne Pollard

That might explain it, I guess. So if I see something before me that I
am inclined to refer to as a "coffee cup", how am I to tell whether or
not it actually _is_ a coffee cup in some essential way?

In fact, it could be your doughnut.

(Old topology joke.)

My doughnuts are generally not continuously deformable to coffee cups.
I'd have to poke them with a stick to make that true, and that would
squirt jam everywhere.