Some updates: Rhyno, a quite successful math major, is correct. I did make an assumption of continuity, which, though warranted, is easily demonstrated through proof of differentiability. Thank you for adding to the proofiness of my proof (with a vocabulary like mine, I'm gonna be the next Colbert).

And thank you, Katharine, for pointing out the VERY interesting fact that the minimum value for f(x) = x^x (between x=0 and x=1) occurs at x = 1/e. I wonder why that is. How did you figure that out?

I'm starting to wonder, even if there are infinite pairs of x and y such that x^x = y^y and x=/=y, how many of those pairs are rational? Is there a finite number of rational values for which it holds true? What is that number? How does one find them?