Local generalizations of the derivatives on the real line

•The article establishes the properties of the generalized (g-) derivatives on the real line.•Derivatives are generalized in terms of the class of the modulus of continuity of the primitive function.•Strictly sub-additive moduli induce g-derivatives which do not vanish only on totally-disconnected subsets of the real line.•Non-strictly sub-additive and super-additive moduli induce ordinary derivatives. From a physical perspective, derivatives can be viewed as mathematical idealizations of the linear growth as expressed by the Lipschitz condition. On the other hand, non-linear local growth conditions have been also proposed in the literature. The manuscript investigates the general properties of the local generalizations of derivatives assuming the usual topology of the real line. The concept of a derivative is generalized in terms of the local growth condition of the primitive function. These derivatives are called modular derivatives. Furthermore, the conditions of existence of the modular derivatives are established. The conditions for the continuity of the generalized derivative are also demonstrated. Finally, a generalized Taylor–Lagrange property is proven. It is demonstrated that only the Lipschitz condition has the special property that the derivative function is non-trivially continuous so that the derivative does not vanish.