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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
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