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Let *F* be the set of all functions defined on the
interval *(0,1)*. Such a function associates a
unique number with every number *0 < x < 1*. We
do not assume that it has any special properties like being
continuous. (It's a good **exercise** to show that the
set of continuous functions on *(0,1)* is in fact
equivalent to the interval *(0,1)* ) That set is
larger than the set of real numbers in *(0,1)*.

To see this we assume the statement is true and derive a contradiction. So suppose there is a map

r <---> fr (*)

that associates a function fr wit a number *0 < r <
1*, such that for every function *f* there is a
unique *r* in *(0,1)* such that *f=fr*.
Now construct a function *g* which is such that *
g(r)* does not equal *fr(r)*, for all * 0 <
r < 1*. Such a function can be obtained easily, for
example via the definition

g(r):=fr(r)+1.

Then clearly there is no real number *r* that is
associated with *g* and we have a contradiction. An
association (*) does not exist.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[16-Aug-1996]