Context. Studies of the mass function (MF) of open clusters of different ages allow us to probe the efficiency with which brown dwarfs evaporate from clusters to populate the field. Surveys of older clusters (age ≳ 100 Myr) are not affected so severely by several problems encountered in young clusters, such as intra-cluster extinction and large uncertainties in brown dwarf models. Aims: We present the results of a deep photometric survey to study the MF of the central region of the old open cluster Praesepe (age 590+150-120 Myr, distance 190+6.0-5.8 pc), down to the substellar regime. Methods: We performed an optical (riz and Y-band) photometric survey of Praesepe using the Large Binocular Telescope Camera covering an area of 0.59 deg2 in the cluster centre from i ~ 19.0 mag (~100 MJup) down to a 5σ detection limit at i ~ 25.6 mag (~40 MJup). The survey is approximately 95% complete at i = 23.8 mag and z = 22.0 mag (~55 MJup). Results: We identify 59 cluster member candidates, of which 37 are substellar, by comparing with the predictions of a dusty atmosphere model. The MF of those candidates rises from the substellar boundary until ~67 MJup and then declines. This is quite different from the form inferred for other open clusters older than 50 Myr, but seems to be similar to those found in very young open clusters, the MFs of which peak at ~10 MJup. Either Praesepe really does have a different MF from other clusters or they had similar initial MFs but a different dynamical evolution. Since most of the candidates are faint, we lack astrometric or spectroscopic follow-ups to test their memberships. However, the contaminations by field dwarfs, galaxies, or giants are found to have little effect on the shape of MF and therefore the MF of "real" cluster members should have similar characteristics. Tables 2 and 3 are available in electronic form at http://www.aanda.org