I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) &lt; s for all p or f(p) &gt; s for all s.So here's what I know so far:-Because f is continuous and the metric space M is connected, f(M) is connected.-f(M) is connected in the real numbers, so if x, y ϵ f ( M ) and x &lt; z &lt; y, then z ϵ f ( M )-I have a feeling the intermediate value theorem will play a part in this, but I'm not sure how.I'm having trouble relating these three points.

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I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) &amp;lt; s for all p or f(p) &amp;gt; s for all s.So here&#039;s what I know so far:-Because f is continuous and the metric space M is connected, f(M) is connected.-f(M) is connected in the real numbers, so if x,  y  ϵ  f  (  M  ) and x &amp;lt; z &amp;lt; y, then   z  ϵ  f  (  M  )-I have a feeling the intermediate value theorem will play a part in this, but I&#039;m not sure how.I&#039;m having trouble relating these three points.

Lara Alvarez · Accepted Answer

Proof 1: If   f  (  p  )  ≠  s for any   p  ∈  M, then      f          −      1        (  (  s  ,  ∞  )  )  ∪      f          −      1        (  (  −  ∞  ,  s  )  )  =  MThe two sets on the left are open and disjoint. Since   M is connected, one of them is   M and one of them is   ∅. So either      f          −      1        (  (  s  ,  ∞  )  )  =  M    ⟹    ∀  p  ∈  M  ,     f  (  p  )  &amp;gt;  sor      f          −      1        (  (  −  ∞  ,  s  )  )  =  M    ⟹    ∀  p  ∈  M  ,     f  (  p  )  &amp;lt;  sQ.E.D.Proof 2: Suppose it is not true that  (  ∀  p  ∈  M  ,     f  (  p  )  &amp;gt;  s  )   or   (  ∀  q  ∈  M  ,     f  (  q  )  &amp;lt;  s  )Since   s is not a value of   f, we have  (  ∃  p  ∈  M  ,     f  (  p  )  &amp;lt;  s  )   and   (  ∃  q  ∈  M  ,     f  (  q  )  &amp;gt;  s  )By the Intermediate Value Theorem, there exists   r  ∈  M with   f  (  r  )  =  s. This contradicts the condition that   s is not a value of   f. Q.E.D.

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never e

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