The Incremental Potential Contact (IPC) method enables robust complex simulations of deformable objects with contact and friction. The key to IPC's robustness is its strict adherence to geometric constraints, avoiding intersections, which are a common cause of robustness issues in contact mechanics. A key element of the IPC approach to contact is a geometric barrier function, which is defined directly in the discrete setting. While IPC achieves its main goal of providing guarantees for contact constraints, its parameters need to be chosen carefully to avoid significant simulation artifacts and inaccuracies. We present a systematic derivation of an IPC-like continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence of shape, and absence of forces in rest configurations, based on the idea of candidate sets. Our potential is formulated in a way independent of surface discretization. This new potential is suitable for piecewise-linear surfaces and its efficiency is similar to standard IPC. We demonstrate its behavior and compare it to IPC on a range of challenging contact examples.