User blog comment:Edwin Shade/A Proof/@comment-1605058-20171104122735/@comment-30754445-20171109050735

Depends on what you mean by "extension".

If you just want the function to be continuous, then there's the usual way:

a^^b = ab for 0<b<1

a^^b = aa^^(b−1)  for b≥1

This works for any positive real numbers a and b, and similar versions exist for pentation, hexation, etc.

The "catch" is that the above definition doesn't give a smoothly growing curve. It has "growth sprouts" whenever b crosses an integer. The easiest way to see what this means exactly, is to look at an example:

5^^1.97 ~ 2139

5^^1.98 ~ 2422

5^^1.99 ~ 2748

5^^2.00 = 3125

5^^2.01 ~ 3866

See that jump? Just below 2.00, it grows by about 300 or 400 every step. Then, from 2.00 to 2.01, it grows by over 700. Ordinary exponential functions (like 5^x) do not such jumps. And usually, when mathematicians consider the extension of a function to fractional numbers, they require it to grow in a completely smooth way. The technical terms for this is "infinitely differentiable".

As far as I know, there's no known extension of tetration to fractional numbers that is infinitely differentiable.