Question

Function which is not normalizable but acceptable as a wave function

Answer 1

From my limited knowledge of this subject, I would say that a non-normalizable wave-function wouldn't really make any physical sense.

Remember, the wave-function is a function whose value squared evaluated between two points represents the probability that the particle will be found between those two points. So, the restriction that wave functions be normalized is basically just a nod to reality - the particle must be found SOMEWHERE