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?

Question

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?

frethi38 · Accepted Answer

This is not true even for polynomial functions. Consider   A  =            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  ∈  [  0  ,  1  ]  ∖  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  &amp;gt;      v    p    (      b    2    ), then       p    n    ∈  [  0  ,  f  (  b  )  ]  ∖  f  (  A  ).You will have better chance if you work with             C        p  .

Is there a p -adic analogue to the intermediate

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