TITLE: $\alpha$-derivative (concept)
QUESTION [11 upvotes]: I found the following definition: Given an real number $\alpha$, we say that a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is $\alpha$-differentiable at $0$ if exists the limit: $$\lim_{t \to 0^+} \frac{f(t) - f(0)}{t^\alpha}$$
It's not very hard to guess what would be the $\alpha$-derivative on another point, but how could we interprete this? I found this here, question 41, but it's in portuguese. The calculation itself is irrelevant, but the concept behind it got me curious. I mean, could be just a definition made up for the exercise, but we can never be too sure. Thanks in advance!

REPLY [2 votes]: Some new references
[1] F. Ben Addaa J. Cresson, About Non-differentiable Functions, Journal of Mathematical Analysis and Applications, 263, (2), 2001, 721-737
[2] F. Ben Addaa J. Cresson, Fractional differential equations and the Schrödinger equation, Applied Mathematics and Computation, 161 (1), 2005,  323–345
[3]  F. Ben Addaa J. Cresson, Quantum derivatives and the Schrödinger equation,
Chaos, Solitons & Fractals
19 (5), 2004, 1323 – 1334
[4] D. Prodanov, Fractional variation of Hölderian functions, Fractional Calculus and Applied Analysis,  18, (3),  2015, 580–602