[Math] ny nonconstant function that grows (at infinity) slower than all iterations of the (natural) logarithm

Question

Is there any nonconstant function that grows at infinity slower than all iterations of the (natural) logarithm?

Answer 1

Yes, just take a function which is equal to $\log x$ on an initial interval such as $[1,K_1]$, then equal to $\log\log x$ on the interval $(K_1,K_2]$, then $\log\log\log x$ etc. where the values $K_1, K_2, \dots$ are chosen sufficiently large to ensure that the function really does tend to infinity. This can always be done, since each of the iterations of the log function does tend to infinity.

This function will not be continuous, but a similar function could certainly be created which is continuous.

This will produce a function which tends to infinity slower than any (fixed) iteration of $\log$. If you want something even slower, then call my function above $\phi (x)$, and repeat my construction using $\phi\phi(x)$ etc on successive intervals.

If this is still not slow enough, then ...