On the space of regular differentiable functions

V. I. Rodionov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 29 (2004) no. 1, pp. 3-32

Résumé

The concept of regular differentiable function are defined. Any piecewise smooth function are regular differentiable function. At the time a modulus of any continuously differentiable function are regular differentiable function. Any regular differentiable function are Lipschitzian. The space of regular differentiable functions are the closure of the space of piecewise linear functions with respect to Lipschitz norm (or Hölder norm). Any regular differentiable function have one-sided derivatives: the left-side derivative are continuous from the left and the right-side derivative are continuous from the right. The concept of regular derivative are generated by one-sided derivatives. Statements about regular derivatives of arithmetic operations, superposition and total variation of regular differentiable functions are proved.