[Ackermann's function](http://en.wikipedia.org/wiki/Ackermann_function) is defined over integers $x$, $y$, $A(x,y)$,
with conditions for when $x=0$ or $y=0$, and otherwise uses recursive
definitions involving arguments $x-1$ and $y-1$.



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> Is there a natural generalizations of $A(x,y)$ for $x,y \in \mathbb{R}$?
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Perhaps this is well-known to the cognoscenti.
I'd appreciate a pointer.



Ackermann's function over the reals
It seems that we may extend the Ackermann function to the non-negative real plane with a continuous real-valued function obeying the Ackermann recursion.


Specifically, consider the Ackermann function defined on the Wikipedia page to which you link, which is equivalent to the following:
$$A(m,n)=\begin{cases} n+1 &\text{ if }m=0\\ 
 A(m-1,1) &\text{ if }m\geq 1\text{ and }n=0\\
 A(m-1,A(m,n-1)) &\text{ if }m\geq 1\text{ and }n\geq 1 \end{cases}$$
Let us simply define the real-valued $A(m,n)$ on the unit square, that is, for $0\leq m,n\leq 1$, in such a way that it has the right values at the four corners and is continuous and increasing in each coordinate on the square. Now we simply extend the domain of our function by applying the recursive rule, which reduces any value to the values appearing inside an already defined box (and we must define $A(m,n)$ when $0\leq n\leq 1$ so as to continuously extend what is required when $n=0$ and $n=1$). That is, we define the function $A(m,n)$ integer-strip-by-integer-strip by recursion on the greatest integer in $m$, and within any integer strip by recursion on greatest integer in $n$. The value of the function inside any box determined by the integer lattice is the composition of continuous functions, and agrees with the values of the ordinary Ackermann function on the integer points themselves. So the result is a continuous function obeying the Ackermann recursion.


