Smooth Extension and Approximation Theorems

Let \(X\) be any subset of \(\mathbb {R}^n\) and \(f:X\to \mathbb {R}\) a function with the following property. For every \(x\in X\) there is a neighborhood \(U_x\) of \(x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Show that there is an open set \(U\) in \(\mathbb {R}^n\) containing \(X\) and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Prove that for any two disjoint closed subsets \(A,B\subset \mathbb {R}^n\) there is a smooth function \(f:\mathbb {R}^n\to \mathbb {R}\) such that \(f=0\) on \(A\) and \(f=1\) on \(B\).

Let \(X\) be the standard “middle thirds” Cantor set in \(\mathbb {R}\), viewed as a subset of the plane \(\mathbb {R}^2\). Assume that \(f:X\to \mathbb {R}\) is a function with the property that for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb {R}^2\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f_x|_{U_x\cap X}=f|_{U_x\cap X}\). Show that there is an open set \(U\) in \(\mathbb {R}^2\), \(U\supset X\), and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Let \(X\) be any subset of \(\mathbb {R}^n\) and \(f:X\to \mathbb {R}\) a function with the following property. For every \(x\in X\) there is a neighborhood \(U_x\) of \(x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Show that there is an open set \(U\) in \(\mathbb {R}^n\) containing \(X\) and a smooth function \(g:U\to \mathbb {R}\) such that \(g|_X=f\).

Recall that a map \(f:X\to \mathbb {R}\) defined on a subset \(X\subset \mathbb {R}^n\) is said to be smooth if for every \(x\in X\) there is an open set \(U_x\ni x\) in \(\mathbb {R}^n\) and a smooth function \(f_x:U_x\to \mathbb {R}\) such that \(f=f_x\) on \(X\cap U_x\). Prove that if \(f:X\to \mathbb {R}\) is smooth then there is a neighborhood \(U\) of \(X\) and a smooth extension \(\tilde f:U\to \mathbb {R}\).