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Back to MA_598R_pweigel_Summer_2009_Lecture_4 $ \text{True of False: If } f \text{ is a non-negative function defined on } \mathbf{R} \text{ and } $

$ \int_{\mathbf{R}}{f} dx < \infty $

$ \text{then } \lim_{|x|\rightarrow\infty}f(x)=0 $

$ \text{Solution: False. Let } f(x)= \begin{cases} 1 & x\in \mathbf{Z} \\ 0 & \text{otherwise}\end{cases} $

$ \text{then } \int_{\mathbf{R}}{f} dx = 0 \text{, but }\lim_{|x|\rightarrow\infty}f(x) \text{ does not exist.} $

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Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood