MA_598R_pweigel_Summer_2009_Lecture_4
$ \text{4.1) True of False: If } f \text{ is a non-negative function defined on } \mathbb{R} \text{ and } $
$ \int_{\mathbb{R}}{f} dx < \infty $
$ \text{then } \lim_{|x|\rightarrow\infty}f(x)=0 $
$ \text{Solution: False. Let } f(x)= \begin{cases} 1 & x\in \mathbb{Z} \\ 0 & \text{otherwise}\end{cases} $
$ \text{then } \int_{\mathbb{R}}{f} dx = 0 \text{, but }\lim_{|x|\rightarrow\infty}f(x) \text{ does not exist.} $
-Ben Bartle
