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The impending Javalambre Physics of the accelerating universe AstrophysicalSurvey (J-PAS) will be the first wide-field survey of $\gtrsim$ 8500 deg$^2$ toreach the `stage IV' category. Because of the redshift resolution afforded by54 narrow-band filters, J-PAS is particularly suitable for cluster detection inthe 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 absorptionand emission lines. Here we evaluate the cluster selection function for J-PASusing N-body+semi-analytical realistic mock catalogues. We optimally detectclusters from this simulation with the Bayesian Cluster Finder, and we assessthe completeness and purity of cluster detection against the mock data. Theminimum halo mass threshold we find for detections of galaxy clusters andgroups with both $>$80\% completeness and purity is $M_h \sim 5 \times10^{13}M_{\odot}$ up to $z\sim 0.7$. We also model the optical observable,$M^*_{\rm CL}$-halo mass relation, finding a non-evolution with redshift andmain scatter of $\sigma_{M^*_{\rm CL} | M_{\rm h}}\sim 0.14 \,dex$ down to afactor two lower in mass than other planned broad-band stage IV surveys, atleast. 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 $\sigma_{M^*_{\rm CL} | M_{\rmh}}\sim 0.12 \, dex$. Therefore J-PAS will provide the largest sample ofclusters and groups up to $z\sim 0.8$ with a mass calibration accuracycomparable to X-ray data.

An Accurate Cluster Selection Function for the J-PAS Narrow-Band wide-field survey