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finite value

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A rational number r = p/q, where p, q are in Z, is said to be properly reduced if p and q (q > 0) have no common integral factor other than +1 or -1. Define the function f as follows;

f(x) = q , if x = p/q, properly reduced.
f(x) = 0 , if x is irrational.

Prove that for every real number x, f fails to be bounded at x.

Note: f: A -> B is bounded if there exist a real number M > 0 such that |f(x)| <= M for every x in A.

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This post assesses a finite value.

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The statement should be:
Prove that for every real number x, f fails to be bounded at any neighborhood of x.

Proof:

We consider any real number x and any bound M>0 and any neighborhood U(x;e) = {y: |y-x|<e} for some e>0.
We want to find some y in U(x;e), such that f(y)>M.
First, we know that the set of rational ...

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