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About non-differentiable functions. (English) Zbl 0995.26006

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” ${d}^{\alpha }f\left(x\right)$ 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

${d}^{\alpha }f\left(x\right)={\Gamma }\left(1+\alpha \right)\underset{t\to x}{lim}\frac{f\left(t\right)-f\left(x\right)}{{|t-x|}^{\alpha }}\phantom{\rule{2.em}{0ex}}\left(1\right)$

(under the assumption that ${d}^{\alpha }f\left(x\right)$ exists) and its consequences.

Reviewer’s remarks. 1. The construction ${d}^{\alpha }f\left(x\right)$ 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 $f\left(t\right)$ 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

${d}^{\alpha }f\left(x\right)=\frac{1}{{\Gamma }\left(1-\alpha \right)}\underset{t\to x}{lim}\left[\frac{f\left(t\right)-f\left(x\right)}{{|t-x|}^{\alpha }}+\alpha \text{sign}\left(t-x\right){\int }_{x}^{t}\frac{f\left(t\right)-f\left(s\right)}{{|t-s|}^{1+\alpha }}ds\right]\phantom{\rule{2.em}{0ex}}\left(2\right)$

which coincides with (1) for functions $f\left(t\right)$ which behave at a point $x$ as the power function ${|t-x|}^{\alpha }$, since

${\int }_{x}^{t}\frac{{\left(t-x\right)}^{\alpha }-{\left(s-x\right)}^{\alpha }}{{\left(t-s\right)}^{1+\alpha }}ds=B\left(1+\alpha ,-\alpha \right)+\frac{1}{\alpha },\phantom{\rule{1.em}{0ex}}t>x·$

From (2), in particular, it follows that ${d}^{\alpha }f\left(x\right)\equiv 0$ for any function whose continuity modulus $\omega \left(f,\delta \right)$ satisfies the conditions that ${lim}_{\delta \to 0}\frac{\omega \left(f,\delta \right)}{{\delta }^{\alpha }}=0,$ and $\frac{\omega \left(f,\delta \right)}{{\delta }^{1+\alpha }}$ is integrable.

##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 26A27 Nondifferentiability of functions of one real variable; discontinuous derivatives