  The impending Javalambre Physics of the accelerating universe Astrophysical
Survey (J-PAS) will be the first wide-field survey of $\gtrsim$ 8500 deg$^2$ to
reach the `stage IV' category. Because of the redshift resolution afforded by
54 narrow-band filters, J-PAS is particularly suitable for cluster detection in
the range z$<$1. The photometric redshift dispersion is estimated to be only
$\sim 0.003$ with few outliers $\lesssim$ 4\% for galaxies brighter than
$i\sim23$ AB, because of the sensitivity of narrow band imaging to absorption
and emission lines. Here we evaluate the cluster selection function for J-PAS
using N-body+semi-analytical realistic mock catalogues. We optimally detect
clusters from this simulation with the Bayesian Cluster Finder, and we assess
the completeness and purity of cluster detection against the mock data. The
minimum halo mass threshold we find for detections of galaxy clusters and
groups with both $>$80\% completeness and purity is $M_h \sim 5 \times
10^{13}M_{\odot}$ up to $z\sim 0.7$. We also model the optical observable,
$M^*_{\rm CL}$-halo mass relation, finding a non-evolution with redshift and
main scatter of $\sigma_{M^*_{\rm CL} | M_{\rm h}}\sim 0.14 \,dex$ down to a
factor two lower in mass than other planned broad-band stage IV surveys, at
least. For the $M_{\rm h} \sim 1 \times 10^{14}M_{\odot}$ Planck mass limit,
J-PAS will arrive up to $z\sim 0.85$ with a $\sigma_{M^*_{\rm CL} | M_{\rm
h}}\sim 0.12 \, dex$. Therefore J-PAS will provide the largest sample of
clusters and groups up to $z\sim 0.8$ with a mass calibration accuracy
comparable to X-ray data.
