Let $E$ be a Banach space consisting of some kinds of functions from $\mathbf R^n$ to $\mathbf R^m$. Given a two-variable function $f:\mathbf R^m\times\mathbf R^n\to\mathbf R^m$. Suppose that $f\in C(\mathbf R^m;E)$, and for any $y\in\mathbf R^n$, the function $f(\cdot,y):\mathbf R^m\to\mathbf R^m$ is bijective whose inverse is denoted as $g(x,y)=f(\cdot,y)^{-1}(x)$. Suppose also that for any $x\in\mathbf R^m$, $g(x,\cdot)\in E$.
Question: Does $g$ also belong to $C(\mathbf R^m;E)$?
I think the answer is negative in general. But does that hold true for some special cases such as $E=C^k(\mathbf R^n;\mathbf R^m)$ or $E=C^\alpha(\mathbf R^n;\mathbf R^m)$ or $E=L^p(\mathbf R^n;\mathbf R^m)$?
Actually I have no idea to deal with the 'partial' inverse... Could anyone give some hints or comments? TIA...