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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2003 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2003 • Question 4

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 \(g|_X=f\).

Raw LaTeX

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 \(g|_X=f\).

Topic Tags

Smooth Extension and Approximation Theorems