The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map?
A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ sufficiently $C¹$-close to $f$, then there exist a homoemorphism $h$ such that $f∘h=h∘g$. This is the topological conjugacy.
My question is: Why the map $g$ should be of class $C¹$. Why this definition does not works with for example $C⁰$ or a discontinuous map $g$ considered as a small variation of the map $f$.