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About non-differentiable functions. (English) Zbl 0995.26006
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
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