Betragsfunktion

Graph der Betragsfunktion.

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

Graph der Signumfunktion.

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

Es gilt:

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

${\displaystyle \max(x,y)=-\min(-x,-y)}$.

Minimumsfunktion

Es gilt:

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

${\displaystyle \min(x,y)=-\max(-x,-y)}$.