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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 5

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

Topic Tags

Smooth Extension and Approximation Theorems