  We present confusion-limited SCUBA-2 450-$\mu$m observations in the
COSMOS-CANDELS region as part of the JCMT Large Program, SCUBA-2 Ultra Deep
Imaging EAO Survey (STUDIES). Our maps at 450 and 850 $\mu$m cover an area of
450 arcmin$^2$. We achieved instrumental noise levels of
$\sigma_{\mathrm{450}}=$ 0.59 mJy beam$^{-1}$ and $\sigma_{\mathrm{850}}=$ 0.09
mJy beam$^{-1}$ in the deepest area of each map. The corresponding confusion
noise levels are estimated to be 0.65 and 0.36 mJy beam$^{-1}$. Above the 4
(3.5) $\sigma$ threshold, we detected 360 (479) sources at 450 $\mu$m and 237
(314) sources at 850 $\mu$m. We derive the deepest blank-field number counts at
450 $\mu$m, covering the flux-density range of 2 to 43 mJy. These are in
agreement with other SCUBA-2 blank-field and lensing-cluster observations, but
are lower than various model counts. We compare the counts with those in other
fields and find that the field-to-field variance observed at 450 $\mu$m at the
$R=6^\prime$ scale is consistent with Poisson noise, so there is no evidence of
strong 2-D clustering at this scale. Additionally, we derive the integrated
surface brightness at 450 $\mu$m down to 2.1 mJy to be $57.3^{+1.0}_{-6.2}$~Jy
deg$^{-2}$, contributing to (41$\pm$4)\% of the 450-$\mu$m extragalactic
background light (EBL) measured by COBE and Planck. Our results suggest that
the 450-$\mu$m EBL may be fully resolved at $0.08^{+0.09}_{-0.08}$~mJy, which
extremely deep lensing-cluster observations and next-generation submillimeter
instruments with large aperture sizes may be able to achieve.
