$ W^{1,\infty}(D)\ni f:D\to\mathbb R, (x,y)\mapsto f(x,y)$, is a strictly monotonic, $L$-continuous function defined on a convex set $D\subset\mathbb R^2$.
Function $g_c$ is implicitly defined by $f(x,g_c(x))=c$. Given $g_c$ is locally convex on $D$, and $c$ is a constant. What can we imply about the convexity of the function $g_{c+\epsilon}$ within $D$? ($g_{c+\epsilon}$ is also implicitly defined by $f(x,g_{c+\epsilon}(x))=c+\epsilon$).
Can we say that, for any $f$, $\exists a>0$ s.t. $g_{c+\epsilon}$ is also convex $\forall (x,y)\in D'$, $\forall|\epsilon|<a$, and for some $D'\subseteq D$, where the convex set $D'$ can be determined by $\epsilon$ and D?
Intuitively I don't think this will always work for functions on $W^{1,\infty}$. So my other question is, what are the classifications of functions $f$ such that we can always find a set of functions $g_{c+\epsilon}$ is convex given $g_c$ is convex?