I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
$$\lim\_{\varepsilon\to0}\frac{\log(f(x+\varepsilon) - f(x))}{\log(\varepsilon)}.$$


To simplify, lets assume that $f$ is non-decreasing, and of course the limit of $\epsilon$
is from the right. Calling this (me, not Mandelbrot!) the "log-derivative", we can calculate
log-derivative $x^\alpha$ at $x=0$ is $\alpha$ ($\alpha>0$).
For a differentiable function, such that the first derivative is positive, we can calculate
(by Taylor) the log-derivative is $1$, and if the function is almost a constant, such that many derivatives are zero, and k being the least integer such that the $k$th derivative is positive,
the log-derivative is $k$. And so on. The idea of course is that this will be defined, at least for some non-differentiable function. 


My question: Does this procedure have an official name? any references?
It is difficult to google because log-derivative means something else ...



Mandelbrot and "log-derivative"
Suppose $f(x) = \int\_0^x d\mu(t)$ for some measure $\mu$. Then your log-derivative is (almost) equivalent to the quantity $\alpha(x) = \lim\_{r \to 0} \frac{\log \mu(]x-r,x+r[)}{\log r}$. This is known as the *local dimension* of $\mu$ at $x$, and it has been much studied in geometric measure theory. One particular topic I know to be connected with Mandelbrot's work on random measures is called *multifractal analysis*, which refers to the study of the level sets of the local dimension.


