Yes, there are other posts about this, because it is common to slip on this aspect. I think Henryk said it best in this post particularly. I think I have seen something similar in tetration itself as opposed to iter-dec-exp, but I'm not sure how to phrase it.
I first saw it when I was following Galidakis' research with his Puiseux series expansions about log(b). Galidakis uses series about the base b, whereas natural iteration uses series expansions about the iterator t in \( \exp_b^t(x) \). With the linear approximation to tetration, as well as higher approximations, the table that Galidakis gives for the Puiseux series expansions of \( {}^{t}(b) \) will give a discontinuity when the number of times you are differentiating is equal to t. In other words, the function:
Andrew Robbins
I first saw it when I was following Galidakis' research with his Puiseux series expansions about log(b). Galidakis uses series about the base b, whereas natural iteration uses series expansions about the iterator t in \( \exp_b^t(x) \). With the linear approximation to tetration, as well as higher approximations, the table that Galidakis gives for the Puiseux series expansions of \( {}^{t}(b) \) will give a discontinuity when the number of times you are differentiating is equal to t. In other words, the function:
\( (t, k) \mapsto \frac{d}{dt} \left(\frac{d^k}{du^k}\left({}^{t}(e^u)\right)|_{u=0}\right) \)
is continuous and differentiable for all \( t \ne k \). This can only be seen if you already have a continuous extension of tetration to non-integer heights, if not, then its just a bunch of points. This has been bothering me for quite some time now, but I think it may be related to the iterability of \( e^x - 1 \). Since tetration and \( h^x - 1 \) are topologically conjugate, this may help explain this as well...Andrew Robbins
