Consider the following two functions:
\begin{align*} f(x) &= x + 1\\ f(x) &= x - 1 \end{align*}
As these functions increment and decrement $x$, we could call them the $increment$ and $decrement$ functions. I'm not sure if that is the conventional mathematical name for them (nor am I concerned with that). My question is if there is a conventional name used for the following function:
$$\forall x \in \Bbb Z, \ f(x) = \begin{cases} x + 1, &x < 0 \\ 0, & x = 0 \\ x - 1, & x > 0 \end{cases}$$
The only name I could guess at would be a $step$ function since it "steps" a number toward 0.
In signal processing, machine learning, and convex optimization, the function $$ f_\lambda(x) = \begin{cases} x+\lambda & \text{if $x\le-\lambda$,} \\ 0 & \text{if $x\in(-\lambda,\lambda)$,} \\ x-\lambda & \text{if $x\ge\lambda$} \end{cases} $$ is known as soft thresholding or shrinkage by $\lambda$.