  We present a star formation rate function (SFRF) at $z\sim4.5$ based on
photometric data from rest UV to optical of galaxies in the CANDELS GOODS-South
field using spectral energy distribution (SED) fitting. We evaluate the
incompleteness of our sample and correct for it to properly confront the SFRF
in this study with those estimated based on other probes. The SFRF is obtained
down to $\sim10\ M_\odot\ \mathrm{yr}^{-1}$ and it shows a significant excess
to that estimated from UV luminosity function and dust correction based on UV
spectral slope. As compared with the UV-based SFRF, the number density is
larger by $\sim1$ dex at a fixed SFR, or the best-fit Schechter parameter of
$\mathrm{SFR}^*$ is larger by $\sim1$ dex. We extensively examine several
assumptions on SED fitting to see the robustness of our result, and find that
the excess still exist even if the assumptions change such as star formation
histories, dust extinction laws, and one- or two-component model. By
integrating our SFRF to $0.22\ M_\odot\ \mathrm{yr}^{-1}$, the cosmic star
formation rate density at this epoch is calculated to be
$4.53^{+0.94}_{-0.87}\times10^{-2}\ M_\odot\ \mathrm{yr}^{-1}\
\mathrm{Mpc}^{-3}$, which is $\sim0.25$ dex larger than the previous
measurement based on UV observations. We also find that galaxies with intensive
star formation ($>10\ M_\odot\ \mathrm{yr}^{-1}$) occupies most of the cosmic
star formation rate density ($\sim80\%$), suggesting that star formation
activity at this epoch is dominant by intensively star-forming galaxies.

Astrophysics of Galaxies