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I have a function $\phi(x,\omega) \in C^{\infty}(\mathbb{R}^n \times \mathbb{R}^n)$ and I know that $\nabla_{x} \phi \neq 0$ and $h(x,\omega) > 0$, where $$ h(x,\omega) = \det \left( \frac{\partial^2 \phi(x,\omega)}{\partial x_{i} \partial \omega_{j}} \right) $$ I have to show that $\phi(x,\omega) = \phi(x',\omega)$ for every $\omega$ implies $x = x'$. I try to apply an implicit function theorem, but I don't know what to do concretely.

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