A few textbooks seem to use the special example of the vector field

to illustrate that a vector fileld F must be simply connected to be a gradient field.

They also say that

is not a potential function becuase it is not continuously differntiable or because it is not a function at all. I'm having a hard time understanding this. Can someone please explain this in simple terms?

My understanding gets especially blurred when the natural domain of the vector field and the natural domain of the potential function are different. Do you know any other examples like the one above?