The general condition for any function to be periodic is for it to repeat periodically. For a function g() with an independent variable p, and period P0, this means: g(p) = g(p+kP0) for any value of the integer k.
The first part of your answer is just the one I provided and is the answer to the question you actually asked. The second part of your answer is unrelated to the question you actually asked, although I suspected it was what you were probably really after. The question you meant to ask was something like, "What conditions must a time-domain signal meet in order for its Fourier transform to be periodic?" Do you see the somewhat subtle, but critical, difference? Your question only asked what conditions must the Fourier transform meet in order to be periodic, not what conditions a periodic Fourier transform imposes on its inverse transform (i.e., the corresponding time-domain signal).
I'm not sure what "4 periodic" means. In this case I can say that the period is P0 because I defined P0 as the period. In general, however, if you only know that g(p) = g(p+kP0) Then you know that P0 is the period of a subharmonic and it may or may not be the fundamental (which is what is generally implied when we use the term 'period'). In order for P0 to be the fundamental period, you need to add the caveat that P0 takes on the smallest value strictly greater than zero for which this relationship holds.
To rephrase my question is words If g of p is identically 4 then does this make the function g(p) periodic since it conforms to your definition?
Ah, okay. I see what you are saying. I think that, given the clarification I provided for those that like to pick nits (and of which I am frequently one), that this is covered since there does not exist a smallest value of P0 that is strictly greater than zero for which this expression holds.
Excuse me WBahn! I was in a hurry that night I posted the question.I thought people can get my meaning of the question easily.I'm a bit poor in English and from now on should be more careful when posting a comment.
No need to get offended or shout (i.e., use all caps). As I said, the distinction is a bit subtle, and that is all the more the case for a non-native English speaker. That's why I went to the effort to describe the difference between what you asked and what you should have asked -- so that you might learn something that would help improve your English skills. Like it or not, communication is a major part of being an engineer and don't expect that people will get the meaning of what you intended to say as opposed to accepting the meaning of what you actually said. Engineering is not about guessing. So, yes, you need to be more careful in the future.