[UFO] - a contradiction in assuming continuous tetration?

[Gottfried]

The problem is that on the complex plane, the continuum iteration is multivalued. (...)

Indeed, this paradox shows that no matter how we may try to extend \( \mathrm{tet} \) to the \( z \)-plane, we cannot make it injective, at least if our \( \mathrm{tet} \) is continuous (up to a cut, anyway).

Hi Mike -
yes, I think the multivaluedness of the log propagates to the slog, and that this gives problems for the tetrate in the complex plane.
However I can't follow completely. The multivaluedness of the log does not imply, that at each point z the log(exp(z)) is arbitrary; it is multivalued, but the different values are distinct. If we look at a small delta-region around z, the images of log(exp(z+delta)) are continuous around each of the multiple values of log(exp(z)), isn't it? (I mean except of the cut-line). I think, the multivaluedness gives continuous orbits but on distinct pathes, and not, say, continuous "smeared regions" of arbitrary change of direction for some continuous real delta-height.

Hmm - I've near null experience in discussion of such matter, so please bear with me if I'm wrong here or expressed myself unclear.

Gottfried