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Let $ (\mathbf{P},\mathbf{D}) $ (P , D) be a primal-dual pair of SDPs with a nonzero finite duality gap. Under such circumstances, $ \mathbf{P} $ P and $ \mathbf{D} $ 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 (i.e. at the unperturbed data). Nevertheless, under the assumption that both $ \mathbf{P} $ P and $ \mathbf{D} $ D have singularity degree one, we show that a limiting version $ {v_a} $ v a of v is a well-defined monotone decreasing continuous bijective function with domain $ [0, \pi /2] $ [ 0 , π / 2 ] and image $ [v(\mathbf{P}),v(\mathbf{D})] $ [ v (P) , v (D) ] , where $ v(\mathbf{P}) $ v (P) and $ v(\mathbf{D}) $ v (D) are the optimal values of $ \mathbf{P} $ P and $ \mathbf{D} $ D , respectively. The domain $ [0,\pi /2] $ [ 0 , π / 2 ] corresponds to directions of perturbation defined in a certain manner. Thus, $ {v_a} $ 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} $ v a is not continuous. [ABSTRACT FROM AUTHOR]