But no, g(s) is not necessarily equal to 0. I'm not too sure (I hate those limit/convergence theorems), but proving that may be enough (because both terms of the difference converge (*))
Consider the integral
Then divide the integral on intervals which boundaries are integers :
But this is a telescoping sum !
So we have :
And this obviously goes to 0 as N goes to infinity (because )
and this converges