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?
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This is not true even for polynomial functions. Consider $A=\mathbb Z_p=[0,1]$ the unit closed disk and $f(z)=z^2$. Then $f(0)=0, f(1)=1$ but $[0,1]$ is not contained in $f(A)$: $p\in [0,1]\setminus f(A)$.
edit It is not even true that $f(A)$ is a ball for small enough ball $A$. In the above example, take $A=[0,b]$ for any non zero $b$, let $n$ be an odd natural integer such that $n>v_p(b^2)$, then $p^n \in [0, f(b)] \setminus f(A)$.
You will have better chance if you work with $\mathbb C_p$.