The density profiles in a fluid interacting with the two identical solid walls of a closed long slit were calculated for wide ranges of the number of fluid molecules in the slit and temperature by employing density functional theory in the local density approximation. Two potentials, the van der Waals and the Lennard-Jones, were considered for the fluid-fluid and the fluid-walls interactions. It was shown that the density profile corresponding to the stable state of the fluid considerably changes its shape with increasing average density (rho(av)) of the fluid inside the slit, the details of changes being dependent on the selected potential. For the van der Waals potential, a single temperature-dependent critical value rho(sb) of rho(av) was identified, such that for rho(av) < rho(sb) the stable state of the system is described by a symmetric density profile, whereas for rho(av) >= rho(sb) it is described by an asymmetric one. This transition constitutes a spontaneous symmetry breaking of the fluid density distribution in a closed slit with identical walls. For rho(av) >= rho(sb), a metastable state, described by a symmetric density profile, was present in addition to the stable asymmetric one. The shape of the symmetric profile changed suddenly at a value rho(c-h) > rho(sb) of the average density, the density rho(c-h) being almost independent of temperature. Because of the shapes of the profiles before and after the transformation, this transition was named cup-hill transformation. At the transition point, the density of the fluid near the walls decreased suddenly from a liquid-like value becoming comparable with the density of a gaseous phase, and the density in the middle of the slit increased suddenly from a gaseous-like value becoming on the order of the density of a liquid phase. For the Lennard-Jones potential, there are two temperature-dependent critical densities, rho(sb1) and rho(sb2), such that the stable density profile is asymmetric (symmetry breaking occurs) for rho(sb1) <= rho(av) <= rho(sb2) and symmetric for rho(av) outside of the latter interval. These critical densities occur only for temperatures lower than a certain temperature, T-sb,T-0. The cup-hill transition is similar to that for the van der Waals potential at low temperatures but becomes smoother with increasing temperature.