Name for bounded piecewise linear function?

Question

Is there a name for the bounded piecewise linear function

$$ f(x) = \begin{cases} x,& x\in [-1,1] \\ -1,& x < -1, \\ 1,& x > 1. \end{cases} $$

It resembles the tanh function but does not have continuous derivatives.

Answer 1

It is a ramped step function.

You may express $$f(x) = \begin{cases} x,& x\in [-1,1] \\ -1,& x < -1, \\ 1,& x > 1. \end{cases}$$

as a heaviside function.

On the positive reals your function is $$f(t)=t-(t-1)H(t-1)$$ and on the negative reals it is $$g(t)=-f(-t).$$

Answer 2

In the deep learning (artificial neural networks) literature, the function is known as the "hard tanh" function. In their course notes https://cs224d.stanford.edu/lecture_notes/LectureNotes3.pdf for "CS 224D: Deep Learning for NLP", Rohit Mundra and Richard Socher state that "The hard tanh function is sometimes preferred over the tanh function [as an activation function for neural networks] since it is computationally cheaper."