I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel too conceptual to be merely typos. I am strongly thorn between what looks obvious to me, and the what a respectable author, who Grey supposedly is, wrote. I would be grateful for healing the confusion.
Thomae was interested in how functions go to zero. The order of vanishing of the function $f$ at the point $a$, where $f(a) \to 0$, is the rate at which $f(x)$ tends to zero as $x$ tends to $a$. It is clear from drawing graphs that the function $x^2$ goes to zero as x tends to zero faster than $x$, for example. The analysis extends in a straightforward way to rational and even irrational power of $x$, but Thomae noted that the natural logarithm, the function $\ln(x)$, goes very slowly to $-\infty$ as $x$ remains positive and tends to zero, so the function $\frac{1}{\ln (x)}$ goes to zero very slowly as x tends to zero; slower, in fact, than any power of $x$. There are even slower functions, for example $\left( \frac{1}{\ln (x)} \right)^2$ and $\frac{1}{\ln ( \ln (x))}$, and Thomae soon found he had a class of functions on his hands whose orders of vanishing could be called measures, and ‘‘these measures constitute a one-dimensional, continuous manifold (in the sense of Riemann) for the determination of which all our ordinary rational and irrational numbers do not suffice.’’
The problem lies, of course, in the note about "even slower functions". It would seem to me that if $f(x)$ tends to $0$, then $(f(x))^2$ tends to $0$ even "faster", and surely not "slower". Worse still, $\ln ( \ln (x))$ is not even defined for $x < 1$.
I would very much appreciate a sanity check from the community. Apologies if the question is too localised.
This is not the only inaccuracy in Gray's "Ghosts". Thus, on page 62 he includes "continuity" in a list of concepts that Cauchy allegedly defined using "limiting arguments", whereas actually Cauchy defined continuity using infinitesimals and only infinitesimals, on page 34 of his Cours d'Analyse and elsewhere throughout his life (Gray in fact reproduces Cauchy's infinitesimal definition on page 64 in "Ghosts").
More specifically, Cauchy defined a function to be continuous if, in a range of values, an infinitesimal $x$-increment necessarily produces an infinitesimal change of the dependent variable $y$. Calling this a "limit" definition is misleading, as it suggests that Cauchy gave some form of epsilon, delta definition we are familiar with today. This is not the case, and Grabiner did not say anything of the sort, either.