Consider two compact convex sets $C_1, C_2 \subset \mathbb{R}^n$ such that $C_2 \subset C_1$. Let us denote by $\partial C_1$ and $\partial C_2$ their boundaries, that satisfy and $\partial C_1 \cap \partial C_2 = \varnothing$.
Consider two continuous, bounded, functions $f_1: C_1 \rightarrow \mathbb{R}^n$ and $f_2: C_1 \rightarrow \mathbb{R}^n$.
Consider a continuous, bounded, function $f: C_1 \rightarrow \mathbb{R}^n$ such that: $$f(y) = f_1(y) \ \ \forall y \in \partial C_1$$
$$f(y) = f_2(y) \ \ \forall y \in \partial C_2$$
1) Prove that there exists a continuous function $g: C_1 \rightarrow \mathbb{R}_{\geq 0}$ such that:
$$ g(y) = 0 \ \ \forall y \in \partial C_1 $$
$$ g(y) = 1 \ \ \forall y \in \partial C_2 $$
$$ f(x) = ( 1-g(x) ) f_1(x) + g(x) f_2(x) \ \ \forall x \in \text{closure}(C_1 \setminus C_2) $$