$\alpha$-derivative (concept)
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!
Solution 1:
For $0 < \alpha < 1$, this type of derivative at a point "overlooks" (or "squeezes away") rapid--but not too rapid--changes in the values of the function at that point. For example, consider $f(x) = x^{1/3}.$ The ordinary derivative is infinite at $x=0,$ but the $(\frac{1}{3})$-derivative at $x = 0$ is equal to $1$ and at every other point the $(\frac{1}{3})$-derivative of $x^{1/3}$ is zero. Also, if $\frac{1}{3} < \alpha < 1,$ then the $\alpha$-derivative of $x^{1/3}$ is $+\infty$ at $x = 0$ and at every other point the $\alpha$-derivative of $x^{1/3}$ is zero. Finally, if $0 < \alpha < \frac{1}{3},$ then the $\alpha$-derivative of $x^{1/3}$ is zero at every point (including $x = 0$). The analogous results hold if we replace $\frac{1}{3}$ with a fixed number $\beta$ such that $0 < \beta < 1.$ Roughly speaking, the closer $\alpha$ is to $0,$ the more the $\alpha$-derivative "overlooks" sharp spikes in the graph of the function, and the closer $\beta$ is to $0,$ the sharper the spike in the graph of $y = x^{\beta}$ at $x = 0.$ Incidentally, by "spike" I simply mean that the function "becomes vertical" at the point (as $x^{1/3}$ does at $x = 0$), and so the graph may not be "spikey" in the usual sense of the word (like $x^{2/3}$ is at $x = 0$). Nonetheless, I thought the word fit well in roughly visualizing the situation, so I decided to use it.
There are continuous functions $f$ that not only fail to have a finite ordinary derivative at each point, but $f$ has the property that the absolute value of the difference quotients approach infinity at each point. Thus, we might say that the derivative becomes "unsigned infinite" at each point. [It is not possible for a continuous function to have at each point a positively infinite derivative, or to have at each point a negatively infinite derivative.] I believe that all the standard nowhere differentiable continuous functions have this property, but it is not difficult to see that some nowhere differentiable continuous functions can have points at which the absolute value of the difference quotients don't approach infinity. For example, use appropriately constructed nowhere differentiable continuous functions, one defined only for non-negative values of $x$ and the other defined only for non-positive values of $x$ and each of them equal to zero for $x=0,$ that approach the graph of $y = |x|$ sufficiently rapidly as $x \rightarrow 0$ so that the result is a function having at $x=0$ a left derivative of $-1$ and a right derivative of $1.$
However, it is also the case that most of the standard examples of nowhere differentiable continuous functions have, for some $0 < \alpha < 1,$ an $\alpha$-derivative equal to zero at every point. [See, for example, the section Hölder continuity at the [Wikipedia webpage Weierstrass function.] Thus, roughly speaking, there is usually a power function $x^{\beta}$ (for some $\beta$ sufficiently close to $0$) such that the nowhere differentiable continuous function has no spikes that are as spikey as the spike that $x^{\beta}$ has at $x = 0.$ However, it is possible to construct continuous functions that have, for each $0 < \alpha < 1,$ an (unsigned) infinite $\alpha$-derivative at each point. That is, a function that at every point is more spikey than the spikeness of any of the $x^{\beta}$ functions at $x = 0.$ In fact, most continuous functions in the sense of Baire category have this property, a result that was first proved by Auerback/Banach [1] (1931), and later extended in various other ways by Jarník [9] [10] (1933, 1936), Petrův [17] (1958), and others.
Most of the time various Hölder conditions are used in mathematics, but $\alpha$-derivatives (via the $\alpha$-Dini derivates) can be a convenient way to formulate more precise statements about the behavior of a function at a point. Incidentally, for $\alpha > 1,$ the $\alpha$-derivative can be used to classify points at which an ordinary derivative is $0$ (the larger the value of $\alpha$ for which the $\alpha$-derivative is nonzero, the "smoother" the function is).
I've assembled a few references below that make use of $\alpha$-derivatives (some use more general scale functions than the $x^{\alpha}$ functions) and I have included web page links for those that appear to be freely available on the internet.
[1] Herman Auerbach and Stefan Banach, Über die Höldersche bedingung [On the Hölder condition], Studia Mathematica 3 (1931), 180-184.
[2] Abram Samoilovitch Besicovitch, On Lipschitz numbers, Mathematische Zeitschrift 30 (1929), 514-519.
[3] Abram Samoilovitch Besicovitch and Harold Douglas Ursell, Sets of fractional dimensions (V): On dimensional numbers of some continuous curves, Journal of the London Mathematical Society (1) 12 (1937), 18-25.
[4] Michel Bruneau, Sur les fonctions non dérivables de Weierstrass [On the non-differentiable functions of Weierstrass], Bulletin de la Société Mathématique de France 105 #4 (1977), 337-347.
[5] Ákos Császár, Sur les nombres de Lipschitz approximatifs [On approximate Lipschitz numbers], Acta Scientiarum Mathematicarum (Szeged) 12(B) (1950), 211-214.
[6] Ákos Császár, Sur les nombres de Lipschitz généralisés [On generalized Lipschitz numbers], Acta Mathematica Academiae Scientiarum Hungaricae 1 (1950), 277-302.
[7] Johannes de Groot, A system of continuous, mutually non-differentiable functions, Mathematische Zeitschrift 64 (1956), 192-194.
[8] Évariste Galois, Démonstration d'un théorème d'analyse [Proof of a theorem in analysis] (1830), pp. 383-385 in R. Bourgne and J. Azra (editors), Écrits et Mémoires Mathématiques d'Évariste Galois [Mathematical Writings and Memoirs of Évariste Galois], Gauthier-Villars, 1962.
[9] Vojtech Jarník, Über die differenzierbarkeit stetiger funktionen [On the differentiability of real functions], Fundamenta Mathematicae 21 (1933), 48-58.
[10] Vojtech Jarník, Sur une propriété des fonctions continues [On a property of continuous functions], Časopis Pro Pestování Matematiky 65 #2 (1936), 53-63.
[11] Ralph Lent Jeffery, Derived numbers with respect to functions of bounded variation, Transactions of the American Mathematical Society 36 #4 (October 1934), 749-458.
[12] Joseph Liberman, Théorème de Denjoy sur la dérivée d'une fonction arbitraire par rapport à une fonction continue [Theorem of Denjoy on the derivative of an arbitrary function with respect to a continuous function], Matematicheskii Sbornik (N.S.) 9(51) #1 (1941), 221-236.
[13] W. Liu, A generalization of a theorem of Auerbach and Banach (Chinese), Acta Mathematica Sinica 23 (1980), 801-807.
[14] W. Liu, A class of continuous functions. II (English), Kexue Tongbao 25 (1980), 370-373.
[15] W. Liu, A class of continuous functions. III (English), Kexue Tongbao 25 (1980), 631-634.
[16] Z. Lu and C. Wang, On the existence of two kinds of functions, Journal of Hangzhou University 10 #1 (1983), 1-8.
[17] Vladimír Petrův, O symetrické derivaci spojitých funkcí [On the symmetric derivative of continuous functions] (Czech), Časopis Pro Pestování Matematiky 83 #3 (1958), 336-342.
[18] Stanislaw Ruziewicz, Ein beispiel zur Hölderschen bedingung [An example for the Hölder condition], Studia Mathematica 3 (1931), 185-188.
Solution 2:
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