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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.