Total IR LF of ALPINE non-target continuum detections (red boxes and black filled circles). The luminosity bins have a width of 0.5 dex in L_IR, and step through the luminosity range in steps of 0.25 dex. For this reason, the individual bins are not statistically independent. The red filled boxes and error bars indicate the 1σ errors derived through simulations (taking into account the photometric redshift uncertainties). The red solid curve is the best fit modified Schechter function derived through the MCMC analysis (see Sect. 4.3.3), while the grey long-dashed curve represents the best fit (modified Schechter function) to the Herschel PEP+HerMES total IR LF by Gruppioni et al. (2013) (interpolated to the redshift bins considered in this work). The Herschel PEP+HerMES 1/V_max data and error bars (at slightly different redshift intervals) are plotted as grey symbols. The green short-dashed curves represent the SCUBA-2 S2CLS derivation by Koprowski et al. (2017). The blue open squares show the ALMA ASAGAO LFs by Hatsukade et al. (2018). The dark green dashed boxes and downward-pointing arrows are the COLDz CO(2-1) and CO(1-0) LFs and limits by Riechers et al. (2019) at z = 2.4 and 5.8, respectively, converted to L_IR as described in the text. The vertical dotted line shows the ALPINE continuum survey completeness limit in L_IR, computed by considering the nominal 860 μm limiting flux of our survey (0.3 mJy; Béthermin et al. 2020) and the template, among those of the library fitting our SEDs, which provided the brightest luminosity at the redshift of the bin.