User blog comment:Nayuta Ito/generalization of arrow notation/@comment-10429372-20141025122001/@comment-1605058-20141026160134

I just figured out that if tetration were to satisfy the condition you suggested, it is bound to be non-continuous. The reason for that is that under your proposal we necessarilly have, for a>1, a^^(1/k)>a^(1/a). This is because, by induction, we have (a^(1/a))^^ka^\frac{1}{a}>1$$, but a^^0=1, so that function is discontinuous at 0.