Extending tetration to real numbers

I have seen Kneser's method of extending tetration to real numbers, but that doesn't seem to be right as I can't see how 2^^0.5 is not 1.55961... As a starting point, we can see that 1+1=2 and 2x0.5=1. Also 1.4142...x1.4142...=2 and 2^0.5=1.4142. Similarly 1.55961...^1.55961...=2 and 2^^0.5 should be 1.55961... To see integer values for all of them, the smallest number after 4 will be 256 where 256x0.5=128, 256^0.5=16 and as 4^4=256, so 256^^0.5 should be 4 We can get values at 0.5, 1.5, 2.5, ... for all numbers by this method, but there should be some way to make the curve continuous as well. Also a^^b should also satisfy a^(a^^b-1), so 2^^1.5 should be 2^1.55961 which is ≈2.9477... and 256^^1.5 should be 256^4 which is 4294967296 Is there any other way to extend tetration to real numbers