On the one hand, I find this answer quite elegant. On the other, it's kind of cheating in that your expanding the functions domain/range. Finding $f^\frac{1}{2}$ of $f: S \to S$ is a trivial exercise if you are willing to expand the domain. It would be like if I solved this question by creating two parallel lines and mapping the original line to the new one and it to the negative values on the original....

I'm curious as to why my answer was down voted. Since this was asked in the context of computer science, I provided a python implementation. Is there something I am unaware of with regards to placing code in answers?

<p> I'm a high school student and I can safely say that most of my peers just don't get what a function is. The only ones who do seem to have learned from programing. Then again, all the really mathematically talented students in my very small school also program... </p> <p>Functions seem to get slipped in somewhere along the line without a proper introduction, and then it is assumed that students know it from there on in.</p>

Oops. I read "suggestions of avenues to pursue" as an opening for any property... Perhaps you might list the properties you are working with?

Wikipedia's explanation of the heuristics, while explaining the idea behind it (fractional iterate) and giving lots of useful information, doesn't provide a nice interpretation. Similarly with all the other content I found...

I understand that it must be frustrating to see a question that seems too low-level posted. Before posting this question, I tried to do due diligence by researching it and asking several math grad students and a (in industry) PHD (who hadn't heard of it before!). Perhaps you could expand on what qualifies as a `research level math question'? Additionally, thinking about a fractional derivative in the indirect manner you describe seems suboptimal, further defending the validity of asking for a more meaningful definition. (I hadn't heard of it this way before hand, but..)