Suppose that $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous function and that it is invertible in its second argument, i.e. for every $x \in \mathbb{R}$, $f(x,\cdot)$ is invertible with inverse $f^{-1}(x,\cdot)$. Define $g(x,y) := f^{-1}(x,y)$. Is $g$ continuous as a function of $(x,y)$? Does the answer change if the assumptions are strengthened so that $f(x,\cdot)$ is strictly increasing (hence invertible) for each $x \in \mathbb{R}$?
I believe that for any given $x$, $g(x,\cdot)$ is continuous since $f(x,\cdot)$ is invertible on an open set (namely, $\mathbb{R}$). What I do not see is what would be required for continuity of $g(\cdot, y)$ for a given $y$, or for the stronger condition that $g$ is continuous in both arguments.