I recently read this paper on defining the fractional derivative for the Wierstrass function. This seems very interesting since derivatives over fractals are generally not well defined. Yet, this paper seems to good to be true, as it doesn't really elaborate on any possible uses or meanings for this result.
Specifically, I'm having trouble understanding the actual implications of this result. In the paper, it states that for all derivatives less than a certain number, the derivative exists, and that for all derivatives greater than a certain order, the derivative doesn't exist, yet it doesn't elaborate on whether or not any of the existing derivatives have useful properties.
My question is this, does this "derivative" over the Wierstrass function have any real physical meaning? More concretely, can I think of the derivative as measuring the slope of the function, or is this just a mathematical trick without application? (Feel free to present a third option). If it has applications, I'd like to know what they are. If not, what was the point in defining this derivative?
What I already know: I'm not asking about the intuition for the fractional derivative, I'm asking about the intuition/meaning of this specific derivative over this specific fractal. So, I'd appreciate it if you don't link/refer me to other papers that don't specifically talk about intuition for this derivative and fractal.