Math 6510 • Differentiable Manifolds

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 \(\tilde f:U\to \mathbb {R}\).