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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 8

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 8

Let \(M\) be a manifold and \(f:M\to \mathbb {R}\) a continuous function. Show that there is a smooth function \(g:M\to \mathbb {R}\) such that \[ |f(x)-g(x)|<1 \] for every \(x\in M\).

Raw LaTeX

Let \(M\) be a manifold and \(f:M\to\mathbb{R}\) a continuous function. Show that
there is a smooth function \(g:M\to\mathbb{R}\) such that
\[
  |f(x)-g(x)|<1
\]
for every \(x\in M\).

Topic Tags

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