We present the eddington bias corrected Specific Star Formation Rate Function (sSFRF) at different stellar mass scales from a sub-sample of the Sloan Digital Sky Survey Data Release DR7 (SDSS), which is considered complete both in terms of stellar mass (M ⋆ ) and star formation rate (SFR). The above enable us to study qualitatively and quantitatively quenching, the distribution of passive/star-forming galaxies and perform comparisons with the predictions from state-of-the-art cosmological models, within the same M ⋆ and SFR limits. We find that at the low mass end (M ⋆ = 10 9.5 − 10 10 M ⊙ ) the sSFRF is mostly dominated by star-forming objects. However, moving to the two more massive bins (M ⋆ = 10 10 − 10 10.5 M ⊙ and M ⋆ = 10 10.5− 10 11 M ⊙ ) a bi-modality with two peaks emerges. One peak represents the star-forming population, while the other describes a rising passive population. The bi-modal form of the sSFRFs is not reproduced by a range of cosmological simulations (e.g. Illustris, EAGLE, Mufasa, IllustrisTNG) which instead generate mostly the star-forming population, while a bi-modality emerges in others (e.g. L-Galaxies, Shark, Simba). Our findings reflect the need for the employed quenching schemes in stateof-the-art models to be reconsidered, involving prescriptions that allow "quenched galaxies" to retain a small level of SF activity (sSFR = 10 −11 yr −1 -10 −12 yr −1 ) and generate an adequate passive population/bi-modality even at intermediate masses (M ⋆ = 10 10 − 10 10.5 M ⊙ ).