Say we have a $C^{\infty}$ function $f: \mathbb{R} \to \mathbb{R}$ and a point $x_0 \in \mathbb{R}$ such that $f'(x_0) = 0$ but $f''(x_0) \not= 0$.
Then does the Morse lemma guarantee us that there exists a $C^{\infty}$ re-parametrization of the real line, call it $\xi: \mathbb{R} \to \mathbb{R}$ such that, for all points in the open interval $(x_0 - \varepsilon, x_0 + \varepsilon)$ for some $\varepsilon > 0$, $$(f \circ \xi)(x) = f(x_0) \pm (\xi(x))^2 \quad \text{and} \quad \xi(x_0) = 0 \quad ? $$
Uncertainties:
1. Are we guaranteed a re-parametrization that is defined on the entire real line (perhaps by composing with another re-parametrization)? Or can we only assume that $\xi$ is defined on $(x_0 - \varepsilon, x_0 + \varepsilon)$? (E.g. perhaps because $\lim\limits_{x \to x_0 - \varepsilon} \xi(x) = + \infty, \lim\limits_{x \to x_0 + \varepsilon} \xi_x = -\infty$, and this can't be fixed by applying another re-parametrization while still satisfying the conclusions of the Morse Lemma.)
2. Are the conditions given sufficient to guarantee that $f$ is a Morse function, and that the hypotheses of the theorem are satisfied? Are they sufficient at least to guarantee that $f$ is a Morse function when restricted to a sub-manifold which is an open interval about $x_0$?
3. Can this trivial/special/degenerate case also be proved using Taylor's theorem? I.e., do we even need to assume that $f$ is $C^{\infty}$, or can we just use Taylor's theorem and assume it is $C^2$?
Note: There is a similar question here. That question is about the proof of the result, whereas my question is more about what the correct statement is (in an even more special case). In particular, I am still not entirely comfortable with changes of coordinates, e.g. identifying abuses of notation.
That is why I try to make explicit the change of coordinates as yet another function $\mathbb{R} \to \mathbb{R}$, and why I try to choose the simplest possible 1-dimensional manifolds, $\mathbb{R}$ or an open subinterval.
1) The parametrization is certainly local, e.g if $f'(y)$ vanishes then $y \notin (x_0- \varepsilon, x_0 + \varepsilon)$, comparing derivatives.
2) $f$ will be locally a Morse function but it can have degenerate critical points outside $x$, again. So really, your statement as the Morse lemma are local statements.
3) I think you need at least $C^3$, for example imagine a function such that $f''(x_n) = 0$ for some sequence $x_n \to x_0$ and $f''(x_0) \neq 0$. Such function should exists and one can't apply Morse lemma here.