I know that given $B$, a compact subset of $\mathbb{R}^n$, and $f : B \to \mathbb{R}^m$, a continuous injective (one-to-one) function, $f^{-1}$ is continuous on $f(B)$. (This true).

I also know that image $f(X)$ of a connected subset $X$ is connected under a continuous function.

Now let $X$ be a connected (non-compact) subset of $\mathbb{R}^n$, and $f : X \to \mathbb{R}^m$ be a continuous injective (one-to-one) function. I am trying (and struggling) to provide a counterexample in which mapping $f^{-1} : f(X) \mapsto X$ is not continuous on $f(X)$. (A rigorous, parametrized example).

Thank you in advance!


How about this one:

$f:[0,2\pi)\to S^1\subset \mathbb{R}^2$ such that $f(x)=(\mathrm{cos}(x),\mathrm{sin}(x))$

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