Function belongs to L2 and L_infinity, but is not vanishing

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The discussion revolves around the challenge of identifying a time function that belongs to both L2 and L_infinity spaces but does not vanish as t approaches infinity. The user seeks suggestions after two weeks of unsuccessful attempts. Dirichlet's function is proposed but dismissed due to its lack of continuity and measurability, which disqualifies it from being in L2 or L_infinity. The conversation emphasizes the need for a function that meets specific mathematical criteria.

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iCyborg
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I got this open problem from my advisor: find a time function which belongs to L2 and L_infinity, but is not vanishing as t -> infinity. Lp here is the p-norm space of functions http://en.wikipedia.org/wiki/Lp_space#Lp_spaces_2

I've worked on this for two weeks now, and still not found the way to proceed.
Any suggestion, please?

Thanks a lot.
 
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Is there a reason Dirichlet's function wouldn't work? (1 on the rationals, zero elsewhere)
 
Thanks.
But those functions are not continuous and measurable, so neither belong to L2 nor L_infinity. Another point, is it vanishing as the variable tend to infinity?
 

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