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We explore the final fates of massive intermediate-mass stars by computing detailed stellar models from the zero-age main sequence until near the end of the thermally pulsing phase. These super-asymptotic giant branch (super-AGB) and massive AGB star models are in the mass range between 5.0 and 10.0 M⊙ for metallicities spanning the range Z = 0.02-0.0001. We probe the mass limits Mup, Mn and Mmass, the minimum masses for the onset of carbon burning, the formation of a neutron star and the iron core-collapse supernovae, respectively, to constrain the white dwarf/electron-capture supernova (EC-SN) boundary. We provide a theoretical initial-to-final mass relation for the massive and ultra-massive white dwarfs and specify the mass range for the occurrence of hybrid CO(Ne) white dwarfs. We predict EC-SN rates for lower metallicities which are significantly lower than existing values from parametric studies in the literature. We conclude that the EC-SN channel (for single stars and with the critical assumption being the choice of mass-loss rate) is very narrow in initial mass, at most 0.2 M⊙. This implies that between ~2 and 5 per cent of all gravitational collapse supernova are EC-SNe in the metallicity range Z = 0.02-0.0001. With our choice for mass-loss prescription and computed core growth rates, we find, within our metallicity range, that CO cores cannot grow sufficiently massive to undergo a Type 1.5 SN explosion. [ABSTRACT FROM AUTHOR]