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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2006 • Question 1

Question

Math 6510 • Differentiable Manifolds Winter 2006 • Question 1

Let \(X\) be a manifold and \(f:X\to \mathbb {R}^k\) a continuous function. Show that for every \(\epsilon >0\) there is a smooth function \(g:X\to \mathbb {R}^k\) such that \[ \|g(x)-f(x)\|<\epsilon \] for every \(x\in X\).

Raw LaTeX

Let \(X\) be a manifold and \(f:X\to\mathbb{R}^k\) a continuous function. Show that
for every \(\epsilon>0\) there is a smooth function \(g:X\to\mathbb{R}^k\) such that
\[
  \|g(x)-f(x)\|<\epsilon
\]
for every \(x\in X\).

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds