I was inspired to explore this by a recent post on the math subreddit, which to my knowledge went nowhere.
Consider the families of functions $x^y$ and $y^x$. Given some $y \in \Bbb R$, the roots $x^y - y^x = 0$ can be numerically evaluated and the area of the functions' intersection can be evaluated. To generalize this, let $$f(x) = \max \{y : x^y - y^x = 0, y \not = x\}$$ $$g(x) = \int_x^{f(x)}{(x^y - y^x) dy}$$
Because it involves an exponential, the function's negative values are going to be defined only for integer values of y. $f(1)$ and $f(e)$ are both undefined, though $g(x)$ tends toward infinity and zero in these cases, respectively.
I'd like to explore more of this function's properties, but the necessity of numerical evaluation seems to make it difficult to compute. I don't have any sophisticated software at my disposal, and Wolfram|Alpha doesn't provide a free user enough power to, say, plot this function. I could try to write my own program to rasterize a plot of the function, but I figured I'd ask about it here before going ahead and doing that.
I suppose my question is, can anyone provide more information about this? Has a function similar to this already been explored?