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\subsection{Definitions}  The classical supply and demand curves $S(p,t)$ and $D(p,t)$ (SD) represent respectively the amount of supply and demand that would reveal themselves   if the price were to be set to $p$ at time $t$. In classical Walrasian auctions, the equilibrium price $p_t^*$ is then set to the value that matches both quantities so that $D(p_t^*,t)=S(p_t^*,t)$. This equilibrium is unique provided the curves are strictly monotonous\footnote{By monotonous\endnote{By  definition, or simply by common sense, the demand curve is a decreasing function of the price whereas the supply curve is increasing.}. The supply and demand curves, as well as the resulting equilibrium price, are represented on Figure \ref{fig:SD} (left). In order to define the dynamics of the supply and demand curves, we also introduce the \emph{marginal supply and demand curves} (MSD), on which we will focus in the   rest of this paper. They are defined as the derivative of the SD curves  
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We will now explore the properties of the above solution at time $t= \tau^-$, i.e., just before the next auction, in the two asymptotic limits $\tau \to \infty$, corresponding to very infrequent auctions, and $\tau \to 0$, corresponding to continuous time auctions.