# real analysis – Can a sequence of absolutely continuous functions be rescaled to be equicontinuous?

Given a function $$f: mathbb R to mathbb R$$, we say $$g: mathbb R to mathbb R$$ is an topological rescaling of $$f$$ if $$g = fh$$ for some orientation preserving homeomorphism $$h$$ of $$mathbb R$$.

Given a sequence $$f_n: mathbb R to mathbb R$$ of absolutely continuous functions, do there exist topological rescalings $$g_n$$ of $$f_n$$ (where the choice of $$h$$ is allowed to depend on $$n$$) such that the restricted functions $${g_n}_{lvert(0, 1)}$$ are equicontinuous?