The definition and extraction of objects and their boundaries within an image are essential in many imaging applications. Classically, two approaches are followed. The first considers the image as a sample of a continuous scalar field: boundaries are implicit surfaces in this field; they are often called iso-surfaces. The second considers the image as a digital space with adjacency relations and classifies elements of this space as inside or outside: boundaries are pairs composed of one inside element and one outside element; they are called digital boundaries. In this paper, we show that these two approaches are closely related. This statement holds for arbitrary dimensions. To do so, we propose a local method to construct a continuous analog of a digital boundary. The continuous analog is designed to satisfy properties in the Euclidean space that are similar to the properties of its counterpart in the digital space (e.g., connectedness, closeness, separation). It appears that this continuous analog is indeed a piecewise linear approximation of an iso-(hyper)surface (i.e., a triangulated iso-surface in the three-dimensional case). Furthermore, we derive significant digital boundary properties from its continuous analog using the Jordan-Brouwer separation theorem: new Jordan pairs, new adjacencies between boundary elements, new Jordan triples. We conclude this paper by illustrating the 3D case more precisely. In particular, we show that a digital boundary can be transformed directly into a triangulated iso-surface. The implementation of this transformation and its efficiency are discussed with a comparison with the classical marching-cubes algorithm.