I am thinking of a proposition analogous to the intermediate value theorem for a scalar function on reals:In a topological space ( X , τ ) where X is a connected set, consider a set S ⊆ X and a continuous function f : R → X. For some a , b ∈ R with a &lt; b, the function satisfies f ( a ) ∈ S ∘ ∧ f ( b ) ∈ X ∖ S ¯ Then, there exists c ∈ ( a , b ) such that f ( c ) ∈ ∂ SIs there any theorem useful for proving this proposition? Or any advice will be appreciated.

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I am thinking of a proposition analogous to the intermediate value theorem for a scalar function on reals:In a topological space   (  X  ,  τ  ) where X is a connected set, consider a set   S  ⊆  X and a continuous function   f  :      R    →  X. For some   a  ,  b  ∈      R   with   a  &amp;lt;  b, the function satisfies  f  (  a  )  ∈      S          ∘        ∧  f  (  b  )  ∈  X  ∖            S      ¯      Then, there exists   c  ∈  (  a  ,  b  ) such that  f  (  c  )  ∈  ∂  SIs there any theorem useful for proving this proposition? Or any advice will be appreciated.

Blaze Frank · Accepted Answer

The assumption that   X is connected is irrelevant; what is relevant is that the interval   [  a  ,  b  ] is connected. Suppose no such c existed. That means that for all   x  ∈  [  a  ,  b  ],   f  (  x  ) is either in       S    ∘   or   X  ∖      S    ¯  . That is, the sets   U  =      f          −      1        (      S    ∘    ) and   V  =      f          −      1        (  X  ∖      S    ¯    ) cover   [  a  ,  b  ]. But   U and   V are disjoint open sets, which both have nonempty intersection with   [  a  ,  b  ], so this contradicts the fact that   [  a  ,  b  ] is connected.

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