User blog comment:Edwin Shade/A Proof/@comment-1605058-20171104122735/@comment-1605058-20171109074652

One can make an infinitely differentiable extension by taking a smooth transition function, call it \(g(x)\), and defining \(a\uparrow\uparrow b=g(x+1)\) for \(-1\leq b\leq 0\) and extending by \(a\uparrow\uparrow(b+1)=a^{a\uparrow\uparrow b}\).

An even stronger, and more technical, assumption on a function is that it's analytic, and that can't be done in the above way. Daniel Geisler has done a considerable amount of work in that direction and that has lead him to an analytic extension of tetration, but unfortunately it doesn't necessarily give real values for real arguments and there is a degree of ambiguity involved in a definition (we have to fix a complex fixed point). I may try to give some more details later on if you're interested, but I have to admit I don't understand all of his work.