J. Math. Anal. Appl. 263, No. 2, 721-737 (2001); corrigendum ibid. 408, 409-413 (2013).
K. M. Kolwankar and A. D. Gangal [“Fractional differentiability of nowhere differentiable functions and dimensions”, Chaos, 6, No. 4, 505-513 (1996; Zbl 1055.26504)] introduced a certain “localization” $$d^\alpha f(x)$$ of the Riemann-Liouville fractional derivative to study the local behaviour of nowhere differentiable functions. The authors present a development of this notion, one of the main statements being the formula $d^\alpha f(x)= \Gamma(1+\alpha)\lim_{ t\to x}\frac{f(t)-f(x)}{|t-x|^\alpha} \tag{1}$ (under the assumption that $$d^\alpha f(x)$$ exists) and its consequences.
Reviewer’s remarks. 1. The construction $$d^\alpha f(x)$$ is equal to zero for any nice function, which roughly speaking, behaves locally better than a Hölder function of order $$\lambda> \alpha$$ and is equal to infinity at all points where it has “bad” behaviour, worse than a Hölder function of order $$\lambda< \alpha$$. Therefore, this construction may serve just as a kind of indicator whether a function $$f(t)$$ at the point $$x$$ is better or worse than the power function $$|t-x|^\alpha$$ (as it is in fact interpreted in the recent paper [K. M. Kolwankar and J. Lévy Véhel, “Measuring functions smoothness with local fractional derivatives”, Fract. Calc. Appl. Anal. 4, No. 3, 285-301 (2001)]. It cannot be named as a fractional derivative, even if a local one. 2. From the well-known Marchaud representation for the fractional derivative it follows immediately that $d^\alpha f(x)= \frac{1}{\Gamma(1-\alpha)}\lim_{ t\to x}\left[ \frac{f(t)-f(x)}{|t-x|^\alpha}+ \alpha \text{sign}(t-x)\int_x^t\frac{f(t)-f(s)}{|t-s|^{1+\alpha}}ds\right]\tag{2}$ which coincides with (1) for functions $$f(t)$$ which behave at a point $$x$$ as the power function $$|t-x|^\alpha$$, since $\int_x^t\frac{(t-x)^\alpha - (s-x)^\alpha}{(t-s)^{1+\alpha} } ds = B(1+\alpha,-\alpha)+\frac{1}{\alpha} , \quad t>x.$ From (2), in particular, it follows that $$d^\alpha f(x)\equiv 0$$ for any function whose continuity modulus $$\omega(f,\delta)$$ satisfies the conditions that $$\lim_{\delta\to 0}\frac{\omega(f,\delta)}{\delta^\alpha}=0,$$ and $$\frac{\omega(f,\delta)}{\delta^{1+\alpha}}$$ is integrable.

### MSC:

 26A33 Fractional derivatives and integrals 26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives

Zbl 1055.26504
Full Text:

### References:

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