If a sequence of polynomials is orthogonal with respect to a bilinear form involving derivatives, these are known as Sobolev orthogonal polynomials. In this paper, a particular case of the bilinear form is considered, called the discrete-continuous one, such as that it involves up to \(N \in \mathbb N\) derivatives of the functions, but the first \(N-1\) appear evaluated only at a fixed point \(c \in \mathbb R\). The authors accomplish a thorough study of the algebraic and differential properties of the corresponding Sobolev orthogonal polynomials and of their connection with the standard orthogonal polynomials. In particular, a new characterization of classical polynomials (as the only orthogonal polynomials that for some \(N \in \mathbb N\) have an \(N\)-th primitive satisfying a three-term recurrence relation) is given.