Intervallum (mathematica)

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

Genera intervallorum

Tria genera intervallorum sunt:

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

Finis intervalli aperti etiam infinitas esse potest:

• $[a,\mathrm{\infty }[:=[a,\mathrm{\infty }):=\{x\in ℝ:x\ge a\}$ vel $]a,\mathrm{\infty }[:=(a,\mathrm{\infty }):=\{x\in ℝ:x>a\}$ aut
• $]\mathrm{\infty },b]:=(\mathrm{\infty },b]:=\{x\in ℝ:x\le b\}$ vel $]\mathrm{\infty },b[:=(\mathrm{\infty },b):=\{x\in ℝ:x<b\}$ aut
• $]\mathrm{\infty },\mathrm{\infty }[:=(\mathrm{\infty },\mathrm{\infty }):=ℝ.$

Casus singulares sunt:

• Si $b>a$ valet, haec intervalla sunt vacua: $[b,a]=(a,a)=[a,a)=(a,a]=\{\}=\mathrm{\varnothing }$
• $[a,a]=\{a\}.$

• Copia
• Functio differentiabilis
• Intrgrale definitum
• Intervallum

Bibliographia

• Interval and segment. Encyclopedia of Mathematics. [1]