Is there a simple (non-trivial) example of smooth $f,g:\mathbb{R}\rightarrow \mathbb{R}$ that are topological conjugate via a homeomorphism $h$ (i.e. $f\circ h = h\circ g$) such that $h$ is not $C^1$?
2026-04-05 16:05:57.1775405157
Example of homeomorphic-conjugate maps that is not $C^1$
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Hint: Play around with
$$\begin{cases} f(x) = 0& x\le 0\\ f(x) = e^{-1/x}& x > 0 \end{cases}$$
and
$$\begin{cases} h(x) = x& x\le 0\\ h(x) = 2x & x > 0 \end{cases}$$