\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 \[ d^\alpha f(x)= \Gamma(1+\alpha)\lim_{ t\to x}\frac{f(t)-f(x)}{|t-x|^\alpha} \tag{1} \] (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 \(\lambdax. \] 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.