A Darboux map is a map between topological spaces that sends every connected subset to a connected subset. Every continuous map has this property. For real-valued functions on an interval, it is the intermediate value property, which can hold even for discontinuous functions.
A bijective Darboux map need not have a Darboux inverse in general. The corresponding question for a bijection 
is more restrictive. For 
an injective function with the intermediate value property
is strictly monotone, and its inverse has the same property.
Peter (2026) proposed counterexamples for 
using recurrent tubes. The publicly released manuscript
remained incomplete and the proposed construction had not received independent verification
as of Sep. 7, 2026. The release disclosed a substantial AI role in developing
the proposed argument.
