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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.

J. M. ain't a mathematician
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Meta
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  • Let $x= -0.1$, then $f(x)=0.9$, so it steps away from 0. – Jay Zha Apr 22 '17 at 02:53
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    I understand what you mean. Instead of $x+1$ and $x-1$, consider $\frac x2$ maybe. This reduces the absolute value of the parameter involved (and in short, you can write that function as $sgn(x)\left|\frac x2 \right|$). I don't know if these functions have a name, though. – Sarvesh Ravichandran Iyer Apr 22 '17 at 02:55
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    Do people actually use "increment" and "decrement" for the functions $x\mapsto x+1$ and $x\mapsto x-1$? When I read "increment" and "decrement" I think of operations that are performed in a computer program (or a denotation thereof), which are not at all the same things as the mathematical functions $x\mapsto x+1$ and $x\mapsto x-1.$ – David K Apr 22 '17 at 02:56
  • I don't know what they're called, if anything, in the mathematical world. You are correct, though, that a programmer would usually name them inc and dec for "increment" and "decrement". – Meta Apr 22 '17 at 02:59
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    I doubt there is a special name for this function. I would not call it a step function, because that's usually reserved for functions like $f(x) = 1$ for $ x\geq 0$, $f(x) = 0$ for $x< 0$. – Jair Taylor Apr 22 '17 at 03:00
  • @YujieZha I am only working with integers, so that case does not apply. I guess I should specify that in the question. – Meta Apr 22 '17 at 03:03
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    The first 2 functions are usually called successor and predecessor. They only be increment and decrement under assignment. The most similar named function I know is ${\rm sgn}$, you could write $f(x) = x + {\rm sgn}(x)$. You could call it "increment by ${\rm sgn}$" or so. – DanielV Apr 22 '17 at 03:05
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    Hmm, I guess it's technically a step function according to Wikipedia's definition. But that's a much larger class of functions that doesn't narrow it down. – Jair Taylor Apr 22 '17 at 03:12
  • I think your function is new and interesting. I don't have any applications for it now, but it is nice and symmetric with $f(-x) = -f(x)$ whereas the other two functions are not. – Somos May 16 '17 at 03:40

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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 $\lambda$.

  • Can you write a little something about why this has applications in convex optimization? It sounds pretty cool. – goblin GONE Apr 22 '17 at 17:25
  • @goblin: I guess it would be more accurate to say it appears in applications of convex optimization (performed componentwise, it's the proximal operator of the $L^1$ norm), not specifically in the core theory of convex optimization itself. –  Apr 23 '17 at 02:09
  • To be concrete, it's a proximal operator because $f_\lambda(x)=\operatorname{argmin}_y \frac{1}{2}(y-x)^2 + \lambda |y|$ – p.s. Mar 27 '24 at 18:32