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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?

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