  The metallicity of local galaxies is tightly related not only to stellar
mass, i.e. the mass-metallicity relation, but also to the star formation rate
(SFR) through the so-called fundamental metallicity relation (FMR); more active
galaxies show lower metallicities at fixed mass. Interestingly, high-z galaxies
up to z~2.5 follow the same relation defined by SDSS locally. However,
different shapes have been proposed for local galaxies, and the existence of a
FMR and the role of the SFR has been recently questioned by some authors. In
this paper we first discuss the various parametrizations of this
mass-metallicity-SFR relation that has appeared in the literature to understand
the origin of their different shapes. We then reanalysed data from CALIFA and
SDSS-IV MaNGA surveys, which were used to suggest no dependency of metallicity
on the SFR in local galaxies. Contrary to those claims, we find that those
datasets are instead fully consistent with the predictions, showing the
expected dependency on the SFR at fixed mass. Finally, we analysed those high-z
data whose consistency with the local relation was questioned. While an
internal dependency on the SFR among the subsamples is difficult to detect at
high-z because of the limited dynamic range sampled in the three parameters and
the intrinsic scatter and uncertainties of such small samples, all these
datasets are compatible with the relation defined locally by SDSS galaxies.
This confirms the lack of evolution of the FMR in these data up to z~2.3.
