A classical result of approximation theory states that the modulus of smoothness of a function f (omega(f,delta)), defined by means of the variation functional, converges to 0 as delta tends to 0 from the right if and only if f is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and nonlinear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of phi-variation in the multidimensional frame. In this paper, working with a concept of multidimensional phi-variation introduced in [Angeloni-Vinti, J. Math. Anal. Appl., 2009], we prove that an analogous characterization holds for the multidimensional phi-modulus of smoothness.