Context : The lower limit for the mass of white dwarfs ( WDs ) with C-O core is commonly assumed to be roughly 0.5 M _ { \odot } .
As a consequence , WDs of lower masses are usually identified as He-core remnants .

Aims : However , when the initial mass of the progenitor star is in between 1.8 and 3 M _ { \odot } , 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 } /M _ { \odot } < 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 M _ { \odot } .

Methods : The pre-WD evolution of stars with initial mass of about 2.3 M _ { \odot } , 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 .

Results : We show that the minimum mass for a C-O WD is about 0.33 M _ { \odot } , so that both He and C-O core WDs can exist in the mass range 0.33-0.5 M _ { \odot } .
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 M _ { \odot } star lasts longer and longer as the total mass decreases .
In particular , the M= 0.33 M _ { \odot } 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 M _ { \odot } 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 .

Conclusions :