  From physical perspective, derivatives can be viewed as mathematical
idealizations of the linear growth.
  The linear growth condition has special properties, which make it preferred.
  The manuscript investigates the general properties of the local
generalizations of derivatives assuming the usual topology of the real line.
  The concept of derivative is generalized in terms of the class of the modulus
of continuity of the primitive function.
  This definition is suitable for applications involving continuous but
possibly non-absolutely continuous functions of a real variable.
  The main application of the approach is the generalization of the Lebesgue
monotone differentiation theorem.
  On the second place, the conditions of continuity of generalized derivative
are also demonstrated.
