  Let $({\bf P},{\bf D})$ be a primal-dual pair of SDPs with a nonzero finite
duality gap. Under such circumstances, ${\bf P}$ and ${\bf D}$ are weakly
feasible and if we perturb the problem data to recover strong feasibility, the
(common) optimal value function $v$ as a function of the perturbation is not
well-defined at zero (unperturbed data) since there are ``two different optimal
values'' $v({\bf P})$ and $v({\bf D})$, where $v({\bf P})$ and $v({\bf D})$ are
the optimal values of ${\bf P}$ and ${\bf D}$ respectively. Thus, continuity of
$v$ is lost at zero though $v$ is continuous elsewhere. Nevertheless, we show
that a limiting version ${v_a}$ of $v$ is a well-defined monotone decreasing
continuous bijective function connecting $v({\bf P})$ and $v({\bf D})$ with
domain $[0, \pi/2]$ under the assumption that both ${\bf P}$ and ${\bf D}$ have
singularity degree one. The domain $[0,\pi/2]$ corresponds to directions of
perturbation defined in a certain manner. Thus, ${v_a}$ ``completely fills''
the nonzero duality gap under a mild regularity condition. Our result is tight
in that there exists an instance with singularity degree two for which ${v_a}$
is not continuous.
