This paper discusses the behavior of a renormalized field theory under scale transformations using dimensional regularization. The continuous dimension method allows for a simple calculation of the theory's behavior under scale transformations, expressed through a differential equation related to the Callan-Symanzik equation. The method involves regularizing the theory in a space-time with dimension n ≠ 4, and expressing the theory in terms of poles at n = 4. The coefficients of these poles are determined by the residues of the poles in the Lagrangian, and the theory is made finite by renormalizing the parameters and rescaling the fields. The paper also discusses the role of the unity of mass in the subtraction procedure and the scaling properties of the renormalized parameters. It shows that the scaling behavior of the parameters can be described by differential equations, and that the coefficients of the poles can be determined from the residues of the poles. The paper concludes that the scaling behavior of the theory can be described by simple equations, and that the results are closely related to the Callan-Symanzik equations. The results are applied to specific theories, such as the λφ⁴ theory and Yang-Mills theory, and show that the theory is finite at n = 4 when the bare coupling constant is given in a specific way. The paper also discusses the implications of these results for the renormalization of mass and the behavior of the theory at different scales.This paper discusses the behavior of a renormalized field theory under scale transformations using dimensional regularization. The continuous dimension method allows for a simple calculation of the theory's behavior under scale transformations, expressed through a differential equation related to the Callan-Symanzik equation. The method involves regularizing the theory in a space-time with dimension n ≠ 4, and expressing the theory in terms of poles at n = 4. The coefficients of these poles are determined by the residues of the poles in the Lagrangian, and the theory is made finite by renormalizing the parameters and rescaling the fields. The paper also discusses the role of the unity of mass in the subtraction procedure and the scaling properties of the renormalized parameters. It shows that the scaling behavior of the parameters can be described by differential equations, and that the coefficients of the poles can be determined from the residues of the poles. The paper concludes that the scaling behavior of the theory can be described by simple equations, and that the results are closely related to the Callan-Symanzik equations. The results are applied to specific theories, such as the λφ⁴ theory and Yang-Mills theory, and show that the theory is finite at n = 4 when the bare coupling constant is given in a specific way. The paper also discusses the implications of these results for the renormalization of mass and the behavior of the theory at different scales.