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Whenever I see tetration discussed here, I inevitably see it asserted that there's no consistent continuous definition for tetration. However, it seems to me that

  1. If we restrict ourselves to positive real values, $^{\frac12}x=e^{W(\ln x)}$ is the only consistent definition for $^{\frac12}x$; and
  2. This definition is sufficient to define tetration to any real height, if we further say that $\displaystyle^tx = {\lim_{a \to t}}\; ^ax$. So, what gives? I don't especially think I'm smarter than professional mathematicians, so if there's an inconsistency I'd like to be able to recognize it. Is there?
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marked as duplicate by Simply Beautiful Art, Namaste, Xander Henderson, max_zorn, Jose Arnaldo Bebita-Dris Aug 13 '18 at 3:51

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    $\begingroup$ Could you expand on how you would define ${}^tx$ using only integer values and the fact that $^{\frac12}x=e^{W(\ln x)}$? $\endgroup$ – Omnomnomnom Aug 10 '13 at 18:43
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    $\begingroup$ so $e^{W(W(\ln(x)))}=e^x $ ? Try W|A wolframalpha.com/input/?i=e^[W[W[ln[x]]]] $\endgroup$ – Gottfried Helms Aug 20 '13 at 20:27
  • $\begingroup$ How are we supposed to know this given we can't really evaluate $^{1/2}x$ well? $\endgroup$ – Simply Beautiful Art Dec 31 '15 at 17:03