Nonlinear embeddings of spaces of continuous functions

We provide a nonlinear version of an extension of the classical Holsztyński theorem due to Jarosz (1984) concerning the into isomorphisms of spaces of continuous functions. More precisely, supposing that K and S are locally compact Hausdorff spaces, we prove that if there exists a map T from an extremely regular subspace A of C₀(K) to C₀(S) satisfying {equation*} 1/M \|f-g\|-L ≤ \|T(f)-T(g)\|≤ M \|f-g\|+L\ ∀ f, g ∈ A, {equation*} with 1≤ M²<2 and L≥ 0, then there exist a subset S₀ of S and a proper mapping φ of S₀ onto K. We show that φ is not only continuous in the obvious case when K is compact, but also in the case when M²< 4/3, where S₀ can be taken locally compact. In the Lipschitz case, that is, when $L=0$, and K and S are intervals of ordinal numbers, our result improves some others by Procházka and Sánchez-González (2017) concerning the Lipschitz embeddings between $C(K)$ spaces and also solves a problem raised by them.