Is "Darboux" homotopy trivial?

Question

By Darboux function between two topological spaces I will understand any function that maps connected subsets to connected subsets.

If $f,g$ are two Darboux functions then Darboux homotopy between them is the usual homotopy except it is assumed to be Darboux instead of continuous.

I remember hearing some time ago that for any two topological spaces, every two Darboux functions on them are Darboux homotopic. Is that true?

Answer 1

I will explain the counter example from the comments.

Write $S^0=\partial I=\{0,1\}$. Let $f:S^0\rightarrow S^0$ be the identity and $g:S^0\rightarrow S^0$ the constant map at $1$. Then these two maps are not Darboux homotopic. For $S^0\times I\cong I\sqcup I$, so a Darboux homotopy between them would a Darboux map $$F:I\sqcup I\rightarrow S^0$$ which would necessary spread one summand of its domain over both the disjoint points in its target.

As an answer to the questions raised in comments, I found the paper On c-Homotopies by J. Pawlak, published in Real Analysis Exchange, 21, 1995-96, 424–429.

Pawlak calls Darboux maps and homotopies connected maps and homotopies, or c-maps and c-homotopies for short. He write $f\cong h$ when $f,g$ are c-homotpic c-maps. In the paper just cited he proves as Theorem 2 the following statement

Let $f,h:X→Y$ be connected functions. Then $f\cong h$ if and only if for each component $Z$ of $X$ there exists a connected set $C_Z\subseteq Y$ such that the cardinality of $CZ$ is less than or equal to that of the continuum and $f(Z)\cap C_Z\neq\emptyset\neq h(Z)\cap C_Z$.