  The star formation rate function (SFRF) and specific star formation rate
function (sSFRF) from the observation are impacted by the Eddington bias, due
to the uncertainties on the estimated SFR. We develop a novel method to correct
the Eddington bias and obtained the intrinsic SFRF and sSFRF from the Sloan
Digital Sky Survey Data Release 7. The intrinsic SFRF is in good agreement with
measurements from previous data in the literature that relied on UV SFRs but
its high star-forming end is slightly lower than those IR and radio tracers. We
demonstrate that the intrinsic sSFRF from SDSS has a bi-modal form with the one
peak found at ${\rm sSFR \sim 10^{-9.7} yr^{-1}}$ representing the star-forming
objects while the other peak is found at ${\rm sSFR \sim 10^{-12} yr^{-1}}$
representing the quenched population. Furthermore, we compare our observations
with the predictions from the IllustrisTNG and Illustris simulations and show
that the ``TNG'' model performs much better than its predecessor. However, we
show that the simulated SFRF and cosmic star formation density (CSFRD) of TNG
simulations are highly dependent on resolution, reflecting the limitations of
the model and today state-of-the-art simulations. We demonstrate that the
bi-modal, two peaked sSFRF implied by the SDSS observations does not appear in
TNG regardless of the adopted box-size or resolution. This tension reflects the
need for inclusion of an additional efficient quenching mechanism to the TNG
model.
