We propose a new way of evaluating the regularity of a graph of a function f. Our approach is based on measuring the growth rate of the lengths of less and less regularized versions of f. This leads to a new index, that we call regularization dimension, dim_R . We derive some analytical properties of dim_R and compare it with other fractional dimensions. A statistical estimator is derived, and numerical experiments are performed, which suggest that dim_R may be computed in a robust way. Finally, we apply the regularization dimension to the study of Ethernet traffic.