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?
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...