Some Elementary Functions

Some function "shapes" are common enough to deserve their own names.

• Identity function. $f:X\to X,\ f(x)=x$ for every $x$; denoted $I_X$ or $I$.
• Constant function. For a fixed $c\in Y$, $f:X\to Y,\ f(x)=c$ for all $x\in X$ -- the value never changes. If $c=0$, it is called the zero function (a special case of a constant function).
• Modulus (absolute value) function. $f:R\to R,\ f(x)=|x|$, where

• Signum function. $f:R\to R,\ f(x)=\begin{cases}x/|x| & x\ne0\\ 0 & x=0\end{cases}$, written $\operatorname{sgn}(x)$; it only ever outputs $-1,0,1$.
• Greatest integer (floor) function. $f(x)=\lfloor x\rfloor$, the greatest integer $\le x$; e.g. $\lfloor1\tfrac15\rfloor=1$, $\lfloor7.23\rfloor=7$, $\lfloor-2\tfrac12\rfloor=-3$ (not $-2$ -- rounding is always downward, even for negatives), $\lfloor6\rfloor=6$, $\lfloor-4\rfloor=-4$. …