# Continuous, infinitely differentiable tetration map

We know that the exponential map is infinitely differentiable; is there a tetration map $$TET$$ defined on $$(0,\infty)$$ such that

1) $$TET(x) > TET(y)$$ when $$x > y$$

2) $$TET(e^x) = TET(x) +1$$

3) $$TET(x)$$ is infinitely differentiable on $$(0,\infty)$$

4) $$TET(1) = 0$$

5) (optional) $$TET$$ has a local taylor series at each point

Note here that $$TET(e) = 1, TET(e^e) = 2, TET(e^{e^e}) = 3...$$

H. Kneser. Reelle analytische Lösungen der Gleichung $$\phi(\phi(x)) =\mathrm{e}^x$$ und verwandter Funktionalgleichungen. J. Reine Angew. Math. 187 (1949), 56–67.