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 \[ F = f\circ g |C.\]
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?
Comments
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?