State and prove an intermediate value theorem for functions mapping into the digital line.The digital line is defined as Z with the topology given by the basis elements B ( n ) = { n } , if n odd, and B ( n ) = { n − 1 , n , n + 1 } if n even.I really have no clue, where to start. I have tried just using the regular intermediate value theorem, but I cannot prove it in this case. Can anyone help me?

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State and prove an intermediate value theorem for functions mapping into the digital line.The digital line is defined as       Z   with the topology given by the basis elements   B  (  n  )  =  {  n  }  , if   n odd, and   B  (  n  )  =  {  n  −  1  ,  n  ,  n  +  1  } if   n even.I really have no clue, where to start. I have tried just using the regular intermediate value theorem, but I cannot prove it in this case. Can anyone help me?

Kenny Kramer · Accepted Answer

Theorem. Let   f  :  X  →      Z   be a function from a space   X to the digital line. Suppose   f is continuous on a connected subspace   A of   X. Let   a,  b  ∈  f  (  A  ). Then for any c∈Z such that   c is between   a and   b, there exists an   x  ∈  A such that   c  =  f  (  x  ).To prove the theorem, first show that any connected subspace of       Z   is a set of consecutive integers (use a proof by contradiction to do this). Since   A is connected and   f is continuous on   A,   f  (  A  ) is also connected, and is therefore a set of consecutive integers. Hence, for any   c between   a and   b,   c  ∈  f  (  A  ), which means   c  =  f  (  x  ) for some   x  ∈  A.

State and prove an intermediate value theorem for functions mapping into the digital line.

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