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Let $t \in \mathbb{R},f:\mathbb{R} \rightarrow \mathbb{R}:t \mapsto f(t)$.

$f$ is required to have:

  1. $\displaystyle\lim_{t \rightarrow \infty} f(t) = L$, where $L>0$ (could be $\infty$, so the limit exists, only it could be $\infty$)

  2. $\displaystyle\int_{0}^{\infty} f(t) dt < \infty$

Is that possible to construct such $f$?

Note:

$f$ can be any function continuous or discontinuous.

I found the discussion here which is involving "tent" function but in that example, the limit does not exist.

Note 2: I have edited the title. Thank you for your fedback !!!

Thank you for any answers or comments.

  • 3
    Your title and your body ask opposite questions. Anyway, the answer to your title question is "yes" and the answer to your body question is "no."2012-02-16
  • 0
    Thank you for your comment... I am sorry for the confusion. I have corrected the title. Yes, the correct question in the body of this post...2012-02-16
  • 1
    After a while the function is greater than $L/2$ (if $L$ is finite) or greater than $1$ (if the limit is $+\infty$).2012-02-16
  • 0
    As Qiaochu said, there is no such function.2012-02-16

2 Answers 2