The lower limit for the mass of white dwarfs (WDs) with C-O core is commonly assumed to be roughly 0.5 Msun. As a consequence, WDs of lower masses are usually identified as He-core remnants. However, when the initial mass of the progenitor star is in between 1.8 and 3 Msun, which corresponds to the so called red giant (RGB) phase transition, the mass of the H-exhausted core at the tip of the RGB is 0.3 < M_H/Msun < 0.5. Prompted by this well known result of stellar evolution theory, we investigate the possibility to form C-O WDs with mass M < 0.5 Msun. The pre-WD evolution of stars with initial mass of about 2.3 Msun, undergoing anomalous mass-loss episodes during the RGB phase and leading to the formation of WDs with He-rich or CO-rich cores have been computed. The cooling sequences of the resulting WDs are also described. We show that the minimum mass for a C-O WD is about 0.33 Msun, so that both He and C-O core WDs can exist in the mass range 0.33-0.5 Msun. The models computed for the present paper provide the theoretical tools to indentify the observational counterpart of very low mass remnants with a C-O core among those commonly ascribed to the He-core WD population in the progressively growing sample of observed WDs of low mass. Moreover, we show that the central He-burning phase of the stripped progeny of the 2.3 Msun star lasts longer and longer as the total mass decreases. In particular, the M= 0.33 Msun model takes about 800 Myr to exhausts its central helium, which is more than three time longer than the value of the standard 2.3 Msun star: it is, by far, the longest core-He burning lifetime. Finally, we find the occurrence of gravonuclear instabilities during the He-burning shell phase.
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