# Differentiability Properties of Log-Analytic Functions

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Logarithmic-analytic functions are iterated compositions (from either side) of globally subanalytic functions (i.e. functions definable in the o-minimal structure $\mathbb{R}$ $_a$$_n$ of restricted analytic functions) and the global logarithm. Their definition is kind of hybrid. From the viewpoint of logic, log-analytic functions are definable in the o-minimal expansion $\mathbb{R}$$_a$$_n$$_,$$_e$$_x$$_p$ of $\mathbb{R}$ $_a$$_n$ by the global exponential function; in fact they generate the whole structure $\mathbb{R}$$_a$$_n$$_,$$_e$$_x$$_p$. But from the point of analysis their definition avoids the exponential function and should therefore also not exhibit properties of the function exp(−1/$x$) as flatness or infinite differentiability but not real analyticity. This seems to be obvious. But the problem is that a composition of globally subanaytic functions and the logarithm allows a representation by ’nice’ terms only piecewise. Moreover the ’pieces’ are in general not definable in $\mathbb{R}$ $_a$$_n$ but only in $\mathbb{R}$$_a$$_n$$_,$$_e$$_x$$_p$. And the existing preparation results for log-analytic functions involve functions which are not log-analytic. But by elaborating on the preparation theorems one can identify situations where the preparation can be carried out inside the log-analytic category. And these situations are sufficient to obtain the following results. We show that the derivative of a log-analytic function is log-analytic. We prove that log-analytic functions exhibit strong quasianalytic properties. We establish the parametric version of Tamm’s theorem for log-analytic functions. (Joint work with Andre Opris)