Function belongs to L2 and L_infinity, but is not vanishing

[iCyborg]

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.

 

[RuncibleSpoon]

Is there a reason Dirichlet's function wouldn't work? (1 on the rationals, zero elsewhere)

 

[iCyborg]

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?