Abstract <p> The paper discusses an applicability condition of a cutoff regularization to a fundamental solution of the Laplace operator in the coordinate representation in the Euclidean space of dimension greater than two. To regularize, we consider a deformation of the solution in a sufficiently small ball centered at the origin by cutting off a singular component, and further supplementing it with a continuous function. It is shown that a set of functions satisfying the applicability condition is not empty. As an example, a family of functions is constructed that can be represented by applying a set of averaging operators to the non-regularized solution, and some specific examples are given. Additionally, it is demonstrated that there exist functions that satisfy the condition in a more strict formulation. </p>