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Let $f(x,\theta,\omega):X \ \times \ \Theta \ \times \ \Omega \to \mathbb{R}$, where $X\subset\mathbb{R}, \ \Theta\subset\mathbb{R}, \text{and} \ \Omega\subset\mathbb{R}_{++}$ are intervals, be jointly continuous in its arguments. Here, $\theta$ and $\omega$ are parameters, and we know $f$ is invertible with respect to $x$.

What I would like to know is whether $f^{-1}$ is also jointly continuous in the parameters $\theta$ and $\omega$. Is joint continuity of $f$ in $\theta$ and $\omega$ sufficient?

Previous posts (here and here) focused on finding expressions for the derivative of the inverse with respect to the parameters, and they assumed the inverse was differentiable (hence, relating to my problem, continuous) with respect to the parameters. What I am interested is conditions that ensure the continuity of the inverse in its parameters.

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