Intervallum (mathematica)

Intervallum est copia numerorum realium quae, pro quibuscumque numeris in ea copia, omnes numeros inter illos numeros continet. Ergo ${\displaystyle I}$ est intervallum si ${\displaystyle a,b\in I,a\leq x\leq b\Rightarrow x\in I}$.

Genera intervallorum

Tria genera intervallorum sunt:

• Clausum: ${\displaystyle [a,b]:=\lbrace x\in \mathbb {R} :a\leq x\leq b\rbrace .}$
• Apertum: ${\displaystyle ]a,b[\$ :=(a,b):=\lbrace x\in \mathbb {R} :a<x<b\rbrace .}
• Semi-apertum vel semi-clausum:
  1. ${\displaystyle [a,b[\$ :=[a,b):=\lbrace x\in \mathbb {R} :a\leq x<b\rbrace } vel
  2. ${\displaystyle ]a,b]:=(a,b]:=\lbrace x\in \mathbb {R} :a<x\leq b\rbrace .}$

Finis intervalli aperti etiam infinitas esse potest:

• ${\displaystyle [a,\infty [\$ :=[a,\infty ):=\lbrace x\in \mathbb {R} :x\geq a\rbrace } vel ${\displaystyle ]a,\infty [\$ :=(a,\infty ):=\lbrace x\in \mathbb {R} :x>a\rbrace } aut
• ${\displaystyle ]\infty ,b]:=(\infty ,b]:=\lbrace x\in \mathbb {R} :x\leq b\rbrace }$ vel ${\displaystyle ]\infty ,b[:=(\infty ,b):=\lbrace x\in \mathbb {R} :x<b\rbrace }$ aut
• ${\displaystyle ]\infty ,\infty [\$ :=(\infty ,\infty ):=\mathbb {R} .}

Casus singulares sunt:

• Si ${\displaystyle b>a}$ valet, haec intervalla sunt vacua: ${\displaystyle [b,a]=(a,a)=[a,a)=(a,a]=\lbrace \rbrace =\emptyset }$
• ${\displaystyle [a,a]=\lbrace a\rbrace .}$

Nexus interni

• Copia
• Functio differentiabilis
• Intrgrale definitum
• Intervallum

Bibliographia

• Interval and segment. Encyclopedia of Mathematics.