Is it possible to study functions from $\mathbb{Q}$ to $\mathbb{Q}$ with ordinary calculus ? Obviously with the limitation that $\mathbb{Q}$ is not complete. So much less limits, derivatives and integrals exist; but does it make sense a tangent in $\mathbb{Q^2}$ ?
Study of functions from $\mathbb{Q}$ to $\mathbb{Q}$
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calculus
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1One difficulty is that if $f'(x) = 0$ for all $x \in \mathbb{Q}$, you couldn't conclude that $f$ is constant. – 2012-10-31
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2Also, any intuition on continuity goes out the window. Take the function which is constantly $0$ if $x^2 < 2$ and $1$ otherwise. It has a clear jump at what would be $x = \pm\sqrt{2}$, but is at no rational point discontinuous. – 2012-10-31