f(x) = x^x

So I think the domain would include:

a) all x where x > 0

b) all rational x where x < 0 and the denominator is odd (negative integers and fractions with odd denominators)

c) 0, if you define 0^0 to be 1 or 0. My two calculators give different answers for this.

But would it include negative irrational numbers? Would it include some and not others? Would there be a difference between algebraic and transcendental numbers?