*(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

(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

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

From (2), in particular, it follows that ${d}^{\alpha}f\left(x\right)\equiv 0$ 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.