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1) Construct a continuous function $f$ on $\mathbb{R}$ that is integrable on $\mathbb{R}$ but $\displaystyle\limsup_{x \to \infty} f(x) = \infty$.

I took the function that is equal to $n$ on $[n, n + 1/n^{3})$ and made it continuous by saying that $f$ is the line segment joining $n$ and $n+1$ on $[n + 1/n^{3}, n+1)$. But I am failing to prove this integrable. For this, $\lim f(x) = \infty$ but how do you prove in general, if limit does not exist that $\limsup$ is infinity?

2) Prove that if $f$ is uniformly continuous and integrable on $\mathbb{R}$ we have $\displaystyle\lim_{|x| \to \infty} f(x) = 0$.

Any help is appreciated.

Thanks

  • 0
    nevermind, sorry about this, i just figured it out2011-11-27
  • 2
    you may post answer for your question.2011-11-27
  • 2
    In fact, answering your own question is explicitly encouraged http://blog.stackoverflow.com/2011/07/its-ok-to-ask-and-answer-your-own-questions/2011-11-27

2 Answers 2