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A classical result of approximation theory states that ${lim}_{\delta \to 0}\omega (f,\delta )=0$, where $\omega$ is the modulus of smoothness of $f$ defined by means of the variation functional, 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 non-linear 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...