  We present ultraviolet, optical, and infrared photometry of the afterglow of
the X-ray flash XRF 050416A taken between approximately 100 seconds and 36 days
after the burst. We find an intrinsic spectral slope between 1930 and 22,200
Angstrom of beta = -1.14 +/- 0.20 and a decay rate of alpha = -0.86 +/- 0.15.
There is no evidence for a change in the decay rate between approximately 0.7
and 4.7 days after the burst. Our data implies that there is no spectral break
between the optical and X-ray bands between 0.7 and 4.7 days after the burst,
and is consistent with the cooling break being redward of the K_s band (22,200
Angstrom) at 0.7 days. The combined ultraviolet/optical/infrared spectral
energy distribution shows no evidence for a significant amount of extinction in
the host galaxy along the line of sight to XRF 050416A. Our data suggest that
the extragalactic extinction along the line of sight to the burst is only
approximately A_V = 0.2 mag, which is significantly less than the extinction
expected from the hydrogen column density inferred from $X$-ray observations of
XRF 050416A assuming a dust-to-gas ratio similar to what is found for the Milky
Way. The observed extinction, however, is consistent with the dust-to-gas ratio
seen in the Small Magellanic Cloud. We suggest that XRF 050416A may have a
two-component jet similar to what has been proposed for GRB 030329. If this is
the case the lack of an observed jet break between 0.7 and 42 days is an
illusion due to emission from the wide jet dominating the afterglow after
approximately 1.5 days.
