Is there an intermediate value like theorem for C ? I know C isn't ordeblack, but if we have a function f : C → C that's continuous, what can we conclude about it?Also, if we have a function, g : C → R ,continuous with g ( x ) &gt; 0 &gt; g ( y ) does that imply a z "between" them satisfying g ( z ) = 0.Edit: I apologize if the question is vague and confusing. I really want to ask for which definition of between, (for example maybe the part of the plane dividing the points), and with relaxed definition on intermediateness for the first part, can we prove any such results?

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Is there an intermediate value like theorem for       C  ? I know       C   isn&#039;t ordeblack, but if we have a function   f  :      C    →      C   that&#039;s continuous, what can we conclude about it?Also, if we have a function,   g  :      C    →      R   ,continuous with   g  (  x  )  &amp;gt;  0  &amp;gt;  g  (  y  ) does that imply a z &quot;between&quot; them satisfying   g  (  z  )  =  0.Edit: I apologize if the question is vague and confusing. I really want to ask for which definition of between, (for example maybe the part of the plane dividing the points), and with relaxed definition on intermediateness for the first part, can we prove any such results?

Kennedi Lawson · Accepted Answer

We do have something similar, it is the fact that continuous image of connected sets are connected. In any topological space, such as       C   with euclidean metric, if   X  ⊂      C   is connected and   f continuous, then   f  (  X  ) is connected. Further if   X is path-connected then so is   f  (  X  ).This is what you mean by &quot;in between&quot;. If you have path connected domain, then if   a,  bb are in the range, then there is a parametrized curve between   a and   b. In       C   it means you can draw a curve from   a to   b, with all the values along the curve also in the range. You also realize if your range is subset of       R  , then your curve will just result in an interval because there is only one dimension for your curve to go from   a to   b, that is it has to take all points between them.Intermediate value theorem is a special case of the theorem that continuous image of path connected sets are path connected. In       R   path connected sets are nothing but intervals.

kakvoglq · Accepted Answer

Consider   f  (  x  )  =      e          π      i      x      .We know that   f  (  0  )  =  1  ,  f  (  1  )  =  −  1.But for no real value   r between 0 and 1 is   f  (  r  )  =  0, or even real valued.Think about how this is a &#039;counter-example&#039;, and what aspect of       C   did we use. It could be useful to trace out this graph.

Is there an intermediate value like theorem for C? I know C isn't ordeblack, but if we have a function f:C->C that's continuous, what can we conclude about it? Also, if we have a function, g:C->R ,continuous with g(x)>0>g(y) does that imply a z "between" them satisfying g(z)=0.

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