I was thinking about tetration of real numbers and was wondering how other operators were extended to the reals.

I'm assuming that addition is defined on the reals by definition of the reals. In other words, 1/2 is the number such that 1/2 + 1/2 = 1, etc. (If I'm wrong, how's this done?)

If addition is defined on the reals, then multiplication can be defined as:

x*a = x + x*(a-1)

x*1 = x

x*0 = 0

How is this definition extended to the reals? For example, how can x*1/2 be defined using only addition? (Edit: And limits, I guess...)

For exponentiation the definition is:

x^a = x * x^(a-1)

x^1 = x

x^0 = 1

Similarly for tetration:

x^^a = x ^ (x ^^ (a-1))

x^^1 = x

x^^0 = 1

So how was x^(1/2) defined? According to Wikipedia, nothing tried so far has been able to define x^^(1/2) in a satisfactory way.

Obviously, the limit lim(e->0), x^^(n+e) = x^^(n-e) should hold, but what other properties should a good solution satisfy? (Wikipedia lists monotonically increasing and continuous. Is that limit the same as continuous?)