I think I can easily prove the seemingly very basic result below. For a paper, I would like to be able to cite a reference for it, rather than having to either write out a proof [which would be cumbersome and space-consuming] or assert it as if it is well-known/obvious.
In the following, for real-valued functions on a subset of $\mathbb{R}^2$, $\partial_1$ and $\partial_2$ denote the partial differential operator with respect to the first and second input respectively.
Proposition (1-D singular IFT). Let $U \subset \mathbb{R}^2$ be a neighbourhood of $(0,0)$, and let $F \colon U \to \mathbb{R}$ be a continuous function such that
- $F(0,0)=0$;
- $\partial_1F$ and $\partial_1^2F$ exist and are continuous on a neighbourhood of $(0,0)$;
- $|\partial_1F(0,x)||x|^{-\frac{1}{2}} \to 0\ $ as $x \to 0$;
- $\partial_1^2F(0,0) < 0$;
- $\partial_2F(0,0)$ exists, with $\partial_2F(0,0)>0$.
Then there exist $\delta,\Delta>0$ and continuous functions $y \colon (0,\delta) \to (-\Delta,0)$ and $z \colon (0,\delta) \to (0,\Delta)$ such that the following statements hold:
- the only zero of $F$ in $(-\Delta,\Delta) \times (-\delta,0]$ is $(0,0)$;
- for each $x \in (0,\delta)$, the only zeros of $F(\,\cdot\,,x)$ in $(-\Delta,\Delta)$ are $y(x)$ and $z(x)$;
- $y(x)x^{-\frac{1}{2}} \to -\sqrt{-2\partial_1^2F(0,0)^{-1}\partial_2F(0,0)}\ $ as $x \to 0+$;
- $z(x)x^{-\frac{1}{2}} \to +\sqrt{-2\partial_1^2F(0,0)^{-1}\partial_2F(0,0)}\ $ as $x \to 0+$;
- $\partial_1F(y(x),x)x^{-\frac{1}{2}} \to +\sqrt{-2\partial_1^2F(0,0)\partial_2F(0,0)}\ $ as $x \to 0+$;
- $\partial_1F(z(x),x)x^{-\frac{1}{2}} \to -\sqrt{-2\partial_1^2F(0,0)\partial_2F(0,0)}\ $ as $x \to 0+$.
(Of course, analogous results for the other three combinations of $\mathrm{sgn}(\partial_1^2F(0,0))$ and $\mathrm{sgn}(\partial_2F(0,0))$ can be formulated.)
Is there any reference with the above result? (I'm happy for the reference to assume stronger smoothness assumptions on $F$.)