Let \(\mathbb R\) be equipped with Lebesgue measure. Suppose \(f\in L^1(\mathbb R)\) is uniformly continuous. Prove that \(\lim _{|x|\to \infty }f(x)=0\).
Question
Math 6210 • Real Analysis
Spring 2011 • Question 4
Question
Spring 2011 • Question 4
Let \(\mathbb R\) be equipped with Lebesgue measure. Suppose \(f\in L^1(\mathbb R)\) is uniformly continuous. Prove that \(\lim _{|x|\to \infty }f(x)=0\).