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The Debye-H\"uckel theory describes rigorously the thermal equilibrium ofclassical Coulomb fluids in the high-temperature $\beta\to 0$ regime ($\beta$denotes the inverse temperature). It is generally believed that theDebye-H\"uckel theory and the systematic high-temperature expansion provide anadequate description also in the region of small {\em strictly positive} valuesof $\beta>0$. This hypothesis is tested in the present paper on atwo-dimensional Coulomb gas of pointlike $+/-$ unit charges interacting via alogarithmic potential which is equivalent to an integrable sine-Gordon fieldmodel. In particular, we apply a form factor method to obtain the exactasymptotic large-distance behavior of particle correlation functions,considered in the charge and number density combinations. We first determinethe general forms of the leading and subleading asymptotic terms at strictlypositive $\beta>0$ and then evaluate their high-temperature $\beta\to 0$ forms.In the case of the {\em charge} correlation function, the leading asymptoticterm at a strictly positive $\beta>0$ is also the leading one in thehigh-temperature $\beta\to 0$ regime. On the contrary, the $\beta\to 0$behavior of the {\em number density} correlation function is accompanied by aninterference between the first two asymptotic terms. Consequently, thelarge-distance behavior of this function exhibits a discontinuity when goingfrom strictly positive values of $\beta>0$ to the Debye-H\"uckel limit$\beta\to 0$. This is the crucial conclusion of the paper: the large-distanceasymptotics and the high-temperature limit do not commute for the densitycorrelation function of the two-dimensional Coulomb gas.

A Trickiness of the High-Temperature Limit for Number Density Correlation Functions in Classical Coulomb Fluids