Is there a $p$-adic analogue to the intermediate value theorem? I know there is a notion of convex sets in the $p$-adic context but can we hope for an intermediate value theorem in this context?
$p$-adic intermediate value theorem
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p-adic-number-theory
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2Could you explain to the uninformed like me what a $p$-adic convex set is? Also: isn't the proof of the intermediate value theorem an argument that ultimately boils down to connectivity? Since $p$-adic things tend to be totally disconnected, I'm wondering what kind of statement you're looking for. – 2011-11-05
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0Let $a$ and $b$ be in $\mathbb{Q}_p$. Then an interval $[a,b]$ is the smallest ball containing both $a$ and $b$. A convex set is a set which for every points $a,b$ in it $[a,b]$ is contained in the set. I am looking for a statement like: if $A \subset \mathbb{Q}_p$ is convex, then $f(A)$ is convex. Clearly this is false in general, but if $f$ is $C^1$ then for sufficiently small balls $A$ $f(A)$ is also a ball, so it is convex. – 2011-11-05