# On the Approximation of the Generalized Cut Function of Degree p+1 By Smooth Sigmoid Functions

## Keywords:

Sigmoid Functions, Cut Function, Generalized Cut Function of Degree P 1, Step Function, Logistic Function, Shifted Logistic Function, Uniform and Hausdorff Approximation## Abstract

We introduce a modification of the familiar cut function byreplacing the linear part in its definition by a polynomial of degree p + 1

obtaining thus a sigmoid function called generalized cut function of degree

p + 1 (GCFP). We then study the uniform approximation of the (GCFP)

by smooth sigmoid functions such as the logistic and the shifted logistic

functions. The limiting case of the interval-valued Heaviside step function

is also discussed which imposes the use of Hausdorff metric. Numerical

examples are presented using CAS MATHEMATICA.