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Is there an intermediate value like theorem for $\mathbb{C}$? I know $\mathbb {C}$ isn't ordered, but if we have a function $f:\mathbb{C}\to\mathbb{C}$ that's continuous, what can we conclude about it?

Also, if we have a function, $g:\mathbb{C}\to\mathbb{R}$ ,continuous with $g(x)>0>g(y)$ 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?

Ishan Banerjee
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  • See https://math.stackexchange.com/questions/1070472/is-this-two-dimensional-version-of-the-intermediate-value-theorem-correct – mr_e_man Nov 17 '23 at 01:29

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We do have something similar, it is the fact that continuous image of connected sets are connected. In any topological space, such as $\mathbb{C}$ with euclidean metric, if $X\subset \mathbb{C}$ 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 "in between". If you have path connected domain, then if $a,b$ are in the range, then there is a parametrized curve between $a$ and $b$. In $\mathbb{C}$ 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 $\mathbb{R}$, 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 $\mathbb{R}$ path connected sets are nothing but intervals.

mez
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Consider $f(x) = e^{\pi i x }$.

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 'counter-example', and what aspect of $\mathbb{C}$ did we use. It could be useful to trace out this graph.

Calvin Lin
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    You seem to intend $f(x) = \exp(\pi i x), f(0) = 1, f(1)=-1$; otherwise take $g(x) = x^2, g(-1)=g(1)$. But for no value $-1 < x< 1$ one has $g(x) = 1$. – Lord_Farin Apr 19 '13 at 14:40
  • @Lord_Farin Agreed. I thought this was cooler, then I realized that endpoints being the same tells us nothing. – Calvin Lin Apr 19 '13 at 14:43
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    @CalvinLin I get it but my question was really a bit vaguer than this. I intended to ask about what definition of "between" would make it work, and whether it could work at all with maybe more relaxed conditions on intermediateness. like with $\Re(f)$ and $\Im(f)$ separarately. – Ishan Banerjee Apr 19 '13 at 14:49
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One way of phrasing the intermediate value theorem is as follows:

(1) Let $f:[-1,1] \to \Bbb{R}$ be continuous. Suppose that $f(-1)=-1$ and $f(1)=1$. Then there there is some $a \in [-1, 1]$ with $f(a)=0$.

By regarding $[-1, 1]$ as the "unit ball" in $\Bbb{R}$, and the set $\{-1, 1\}$ as its boundary the "unit sphere", you can write down an analogous statement for $\Bbb{R}^2 \cong \Bbb{C}$:

(2) Let $f: D \to \Bbb{C}$ be continuous, where $D$ is the closed unit disc in $\Bbb{C}$. Suppose that $f(z)=z$ whenever $|z|=1$. Then there is some $a \in D$ with $f(a)=0$.

This is also a true statement. Like the intermediate value theorem, it's physically very intuitive if you think about it in the right way. One physical version of (2) might be:

(2') If there's a laser pointer shining down the middle of a pipe, and the end of the pipe is covered by an opaque balloon, there's no way of stretching the balloon to reveal the laser pointer (without taking it off the pipe or tearing it).

It is, however, somewhat harder to prove than the intermediate value theorem. It's equivalent to the two-dimensional version of the Brouwer fixed-point theorem, a famous theorem in algebraic/combinatorial topology (in fact, all the proofs of the Brouwer fixed-point theorem that I know of use something like (2) as a key lemma).

Micah
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A book “First concepts of topology” by W.G. Chinn and N.E. Steenrod is devoted to intermediate value theorems for the line and the plane and their applications. The exposition is very intuitive, detailed, and illustrated. A counterpart of the intermediate value theorem for $\Bbb C$ is the main theorem of Part II [p. 86].

Let $f:D\to P$ be a mapping of a disk into the plane, let $C$ be the boundary circle of $D$, and let $y$ be a point of the plane not on $fC$. If the winding number of $f|C$ about $y$ is not zero, then $y\in fD$; i.e. there is a point $x\in D$ such that $fx=y$.

Alex Ravsky
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