Ok, so you mean: Let $S_{D} , S_{D_{\pm}}, S_{D_{[a,b]}}$ be the solution sets of the ODE, the implied/one sided ODE and the subODE. We knew that $S_{D_{\pm}} \supset S_{D}$ and $S_{D_{[a,b]}}\supset S_{D}$, but there may be nontrivial intersection $S_{D_{[a,b]}}\cap S_{D_{\pm}}$
and you are interested in only the continuous functions (in the usual sense) within these sets on whether $S_{D}\vert_{\mathcal{C}}=S_{D_{\pm}}\vert_{\mathcal{C}}$