# Does an absolutely integrable function tend to $0$ as its argument tends to infinity?

Suppose that $f:[0,\infty)\rightarrow\mathbb{R}$ is continuous. Is it true that

$$\int_{0}^\infty|f(t)|dt<\infty\Rightarrow \lim_{t\rightarrow\infty}f(t)=0?$$

If so can you provide a proof, otherwise a counter example. Thank you.

• Not necessarily. Think of a function whose graph consists of "spikes" of height one, centered on the integers, whose widths become so small that the function is absolutely integrable. If the function is uniformly continuous, then it is true. – David Mitra Apr 23 '13 at 13:07
• @DavidMitra, i think that qualifies as an answer and would not leave this question unanswered. – user45099 Apr 23 '13 at 13:40

## 1 Answer

Not necessarily. Think of a function whose graph consists of an infinite number of "spikes" of height one, centered on the integers, whose widths become so small that the function is absolutely integrable

So, for example, for $n>1$, let the $n$'th spike consist of the line segments joining the points $(n-1/ n^2, 0 )$, $(n,1)$, and $(n+1/ n^2,0 )$, so that it encompasses an area of $1/n^2$.

But, if the function is uniformly continuous, then it is true. It fact, one only needs the hypothesis that $f$ is (improperly) integrable here to ensure that it has limit zero at infinity. See this post.