Suppose $ F: C \to \mathbb{R}, C$ the Cantor set, has bounded total variation. Is there a homeo $g : [0,1] \to [0,1]$ and a diffeo (Lipschitz, maybe) $f:[0,1] \to \mathbb{R}$ such that \[ F = f\circ g |C.\]

PermalinkSubmitted by bonatti on Sat, 06/17/2017 - 16:15

Maybe the homeomorphism g is not required to be defined on $[0,1]$ but only on $C$. Otherwize, $f\circ g$ is necessarily monotonous, and the answer of the question would be trivially "no". Even with that the question still looks strange: someone is able to state hypotheses making the question pertinent?

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## Maybe the homeomorphism g is

Maybe the homeomorphism g is not required to be defined on $[0,1]$ but only on $C$. Otherwize, $f\circ g$ is necessarily monotonous, and the answer of the question would be trivially "no". Even with that the question still looks strange: someone is able to state hypotheses making the question pertinent?

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