About non-differentiable functions (Q5955062)

\textit{K. M. Kolwankar} and \textit{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 NEWLINE\[NEWLINEd^\alpha f(x)= \Gamma(1+\alpha)\lim_{ t\to x}\frac{f(t)-f(x)}{|t-x|^\alpha} \tag{1}NEWLINE\]NEWLINE (under the assumption that \(d^\alpha f(x)\) exists) and its consequences.NEWLINENEWLINENEWLINEReviewer'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 [\textit{K. M. Kolwankar} and \textit{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 NEWLINE\[NEWLINEd^\alpha f(x)= \frac{1}{\Gamma(1-\alpha)}\lim_{ t\to x}\left[ \frac{f(t)-f(x)}{|t-x|^\alpha}+ \alpha \text{sign}(t-x)\int_x^t\frac{f(t)-f(s)}{|t-s|^{1+\alpha}}ds\right]\tag{2}NEWLINE\]NEWLINE which coincides with (1) for functions \(f(t)\) which behave at a point \(x\) as the power function \(|t-x|^\alpha\), since NEWLINE\[NEWLINE\int_x^t\frac{(t-x)^\alpha - (s-x)^\alpha}{(t-s)^{1+\alpha} } ds = B(1+\alpha,-\alpha)+\frac{1}{\alpha} , \quad t>x.NEWLINE\]NEWLINE 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.