Continuous Intervals

Reillyman

Is it correct to define a continuous interval, for example R, as follows?

[latex] R = \{x: x \in R\} [/latex]

Surely a continuous interval can't be made up of individual points, I asked our lecturer and they reckon it was correct but it still isn't sitting right with me.

Thoughts?

Eliot Rosewater

Are you sure you've that down right? Does [LATEX]R=\{x:x\in R\}[/LATEX] actually makes sense? You're defining R in terms of itself?

Reillyman

It was the example we were given but I agree it's not the best, this should be better;

[latex] (2, 3) = \{x: x\in R,x>2, x<3\} [/latex]

Eliot Rosewater

Ah, I see now. Yes -- it makes sense to write [LATEX](2,3)=\{ x \in \mathbb{R} : 2<x<3 \}[/LATEX]. Taking your perspective, (2,3) is composed of individual points but there are so many of them in there that it becomes continuous, in the same way that water is composed of many individual H2O particles but appears continuous to us. Ultimately, though, I wouldn't see the need to be philosophical about it: mathematically, [LATEX](2,3)=\{ x \in \mathbb{R} : 2<x<3 \}[/LATEX] is completely precise and correct.

FoxT

Cant help, but I LOVE the integral!