Let $D$ be a set of infinitely smooth (in $C^\infty$) functions from $\mathbb{R}^2$ to $\mathbb{R}$ that are strictly increasing in both arguments. Let $C$ be a compact subset of $\mathbb{R}^N$ and $g:\mathbb{R}^2 \times C \to \mathbb{R}$ be some twice-continuously differentiable function strictly increasing in the first two arguments. $C$ is its parameter space.
I want to show that the following property holds for a generic $f\in D$:
For every $k\in \mathbb{R}$, $x \in C$, the set $ \{(a,b) \in \mathbb{R}^2: \ \ g(a,b; x)= f(a,b)=k \} $ is countable.
By `generically' I mean that it holds on an open and dense set of functions $f$ in $D$.
Intuition: The parameter space of $g$ is finite-dimensional. Thus, there are not enough degrees of freedom to align the level curves of $f$ and $g$ along any region of positive length.
Infinite dimentional Thom's transversality theorem seems relevant, but I do not know how to apply it.
