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let $f$:R->N such that given any $x_1$,$x_2$ belong to R such that $x_1 < x_2$ then there exists a $x_3$ belongs to $(x_1,x_2)$ such that f(x_3)> max(f(x_1),f(x_2)). How many such mappings are possible ?

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There are $\aleph_0^\aleph$ such mappings. The upper bound is trivial. For the lower bound, pick any function $f\colon \mathbb{R} \longrightarrow \mathbb{N}$ such that $f(\pm \frac{p}{q}) = q$.

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    I think your exponent should be c, the power of the continuum. Aleph without a subscript is unclear.2010-11-08