I would like to have an inverse (or/and) implicite function theorem for DC-functions. It seems that I have right definitions, but I fail to prove it...

**Definitions:**

Let $h:\mathbb R^n\to\mathbb R$ be a convex function and $x\in\mathbb R^n$, the set of all linear functionals $\ell:\mathbb R^n\to\mathbb R$ such that $$h(y)\ge h(x)+\ell(y-x)$$ is called

**subdifferential**of $h$ at $x$ --- it will be denoted as $\partial_{x}h$. (In general $\partial_{x}h$ is a nonempty bounded convex set)$f:\mathbb R^n\to\mathbb R$ is called

**DC-function**if it is a difference between two convex functions.$F:\mathbb R^n\to\mathbb R^k$ is called

**DC-map**each coordinate function of $F$ is DC.$x\in\mathbb R^n$ is called

**regular value of a DC-function**$f:\mathbb R^n\to\mathbb R$ if of $f=h_1-h_2$ for some convex functions $h_1$ and $h_2$ and $\partial_x h_1 + (-\partial_x h_2)\not\ni0$. Here $+$ denotes*Minkowski sum*and $(-\partial_x h_2)$ is reflection of $\partial_x h_2$ in the origin.$x\in\mathbb R^n$ is called

**regular value of a DC-map**$f:\mathbb R^n\to\mathbb R^k$ if $x$ is a regular value of $\ell\circ F$ for any non-zero linear map $\ell:\mathbb R^k\to\mathbb R$.