Suppose $(\mathcal{X},d_{x})$ and $(\mathcal{Y},d_{y})$ are metric spaces, and suppose that $\{x_{n}\}$ is a sequence in $\mathcal{X}$ and $\{y_{n}\}$ is a sequence in $\mathcal{Y}$ with $y_{n} = g(x_{n})$, $y = g(x)$ and $g:\mathcal{X} \to \mathcal{Y}$ continuous.
My question is:
- If $d_{y}(y_{n},y)\to 0 $ implies $d_{x}(x_{n},x)\to 0$, is it possible to construct a continuous function $f: \overline{\mathbb{R}} \to \overline{\mathbb{R}}$ such that $d_{x}(x_{n},x) \leq f (d_{y}(y_{n},y))$?
- If (1) is not possible, what if $g$ is uniformly continuous?
Further Background: To me, this question is related the existence of a continuous inverse function $g^{-1}$ that maps "close points" in $\mathcal{Y}$ to "close points" in $\mathcal{X}$. To invoke something like the inverse function Theorem seems to require some assumptions on $g$ beyond continuity or uniform continuity (for example, it seems impossible when $\mathcal{X} = \mathbb{R}^{d}$ with $d>1$ and $\mathcal{Y} = \mathbb{R}$). But even if a continuous inverse exists, I am not sure that continuity of $g^{-1}$ necessarily means we can construct a continuous $f: \overline{\mathbb{R}} \to \overline{\mathbb{R}}$ such that $d_{x}(x_{n},x) \leq f (d_{y}(y_{n},y))$. My intuition says that none of these statements are true, but I have had trouble proving it/providing a counterexample.
References are especially welcome!