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## Main Question or Discussion Point

just a cool fact I thought I'd share with anyone who's interested:

The set of real values functions on any interval in R has cardinality at least 2^c.

Pf: Consider characteristic functions defined on the interval, (a,b). (Note: a characteristic function is a function that can be defined on ANY domain and has range {0,1})

Let E be a subset of (a,b), then the characteristic, g(x) function of E over (a,b) i.e.

0 if x is not in E

g(x) =

1 if x is in E

Now for each subset E of (a,b) there corresponds a unique characteristic function defined on (a,b). Hence the set of all subsets of (a,b) and the set of characteristic functions defined on (a,b) are equivalent.

The set of real values functions on any interval in R has cardinality at least 2^c.

Pf: Consider characteristic functions defined on the interval, (a,b). (Note: a characteristic function is a function that can be defined on ANY domain and has range {0,1})

Let E be a subset of (a,b), then the characteristic, g(x) function of E over (a,b) i.e.

0 if x is not in E

g(x) =

1 if x is in E

Now for each subset E of (a,b) there corresponds a unique characteristic function defined on (a,b). Hence the set of all subsets of (a,b) and the set of characteristic functions defined on (a,b) are equivalent.