User blog comment:Nayuta Ito/generalization of arrow notation/@comment-11227630-20141026002048/@comment-1605058-20141026065047

Nayuta's extension isn't even differentiable everywhere, but I believe it's continuous in all arguments.

I agree that the best extensions would have to be infinitely differentiable, but constructing such extensions can be very hard. Daniel Geisler has defined extension of tetration which is meromorphic (complex differentiable in all points of its domain), but the definition is very complicated, and uses power series terms if which aren't all that simple to compute.

It's still possible that there exists some fairly simple integral which would serve the purpose of extending tetration, but, in my honest opinion, I don't think the same will ever happen to up-arrows.