# Formelsammlung Mathematik: Spezielle elementare Funktionen

### Betragsfunktion

Definition. Betragsfunktion.

Für ${\displaystyle x\in \mathbb {R} }$:

${\displaystyle |x|:={\begin{cases}\;\;\,x&{\text{wenn}}\;x\geq 0,\\-x&{\text{wenn}}\;x<0.\end{cases}}}$

Für ${\displaystyle x,y\in \mathbb {R} }$ gilt:

${\displaystyle |x|={\sqrt {x^{2}}}}$,
${\displaystyle |x|=|{-}x|}$,
${\displaystyle |xy|=|x|\,|y|}$,
${\displaystyle y\neq 0\implies \left|{\frac {x}{y}}\right|={\frac {|x|}{|y|}}}$,
${\displaystyle |x+y|\leq |x|+|y|}$,
${\displaystyle |x-y|\geq ||x|-|y||}$,
${\displaystyle |x|=0\iff x=0}$.

Ableitung:

${\displaystyle x\neq 0\implies {\frac {\mathrm {d} }{\mathrm {d} x}}\,|x|=\operatorname {sgn} (x).}$

### Signumfunktion

Definition. Signumfunktion.

Für ${\displaystyle x\in \mathbb {R} }$:

${\displaystyle \operatorname {sgn} (x):={\begin{cases}\;\;\,1&{\text{wenn}}\;x>0,\\\;\;\,0&{\text{wenn}}\;x=0,\\-1&{\text{wenn}}\;x<0.\end{cases}}}$

Für ${\displaystyle x,y\in \mathbb {R} }$ gilt:

${\displaystyle x=\operatorname {sgn} (x)\,|x|}$,
${\displaystyle |x|=\operatorname {sgn} (x)\,x}$,
${\displaystyle \operatorname {sgn} (-x)=-\operatorname {sgn} (x)}$,
${\displaystyle \operatorname {sgn} (xy)=\operatorname {sgn} (x)\operatorname {sgn} (y)}$,
${\displaystyle y\neq 0\implies \operatorname {sgn} {\Big (}{\frac {x}{y}}{\Big )}={\frac {\operatorname {sgn} (x)}{\operatorname {sgn} (y)}}}$,
${\displaystyle x\neq 0\implies \operatorname {sgn} {\Big (}{\frac {1}{x}}{\Big )}={\frac {1}{\operatorname {sgn} (x)}}=\operatorname {sgn} (x)}$,
${\displaystyle x\neq 0\implies \operatorname {sgn} (x)={\frac {x}{|x|}}={\frac {|x|}{x}}}$.

Integral:

${\displaystyle |x|=\int _{0}^{x}\operatorname {sgn} (t)\,\mathrm {d} t.}$

### Maximumsfunktion

Definition. Maximumsfunktion.

Für ${\displaystyle x,y\in \mathbb {R} }$:

${\displaystyle \max(x,y):={\begin{cases}x&{\text{wenn}}\;x\geq y,\\y&{\text{sonst}}.\end{cases}}}$

Es gilt:

${\displaystyle \max(x,y)={\frac {x+y+|x-y|}{2}}}$,
${\displaystyle \max(x,y)=-\min(-x,-y)}$.

### Minimumsfunktion

Definition. Minimumsfunktion.

Für ${\displaystyle x,y\in \mathbb {R} }$:

${\displaystyle \min(x,y):={\begin{cases}x&{\text{wenn}}\;x\leq y,\\y&{\text{sonst}}.\end{cases}}}$

Es gilt:

${\displaystyle \min(x,y)={\frac {x+y-|x-y|}{2}}}$,
${\displaystyle \min(x,y)=-\max(-x,-y)}$.