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Abstract We consider a surveillance-evasion game in an environment with obstacles. In such an environment, a mobile pursuer seeks to maintain visibility with a mobile evader, who tries to hide from the pursuer in the shortest time possible. In this two-player zero-sum game setting, we study the discontinuities of the value of the game near the boundary of the target set, where the players cannot see each other (the non-visibility region). In particular, we describe the transition between the usable part of the boundary of the target (where the value vanishes) and the non-usable part (where the value is positive). We show that the value exhibits different behaviour depending on the regularity of the obstacles. Namely, we prove that the boundary profile is continuous in the case of smooth obstacles and that it exhibits a jump discontinuity when the obstacle contains corners. Moreover, we prove that, in the latter case, there is a semi-permeable barrier emanating from the interface between the usable and the non-usable part of the boundary of the target set.