periodic discrete and continuous time signals

[KruthikaMithra]

I read in a book that discrete time signals are periodic only if 'f' is a rational number. Why doesn't this rule apply for continuous time signals? For a continuous time signal x(t)=Acos(2πft) , let 'T' be fundamental period. Consider x(t+T) = Acos(2πf(t+T)) = Acos(2πft+2πfT)
By periodic property x(t)=x(t+T) . I.e. Acos(2πft)= Acos(2πft+2πfT) . Will this not imply 2πfT=2πk where k is an integer and hence f=k/T a rational number? Plz help!

 

[tgootee]

That is a restriction, for discrete-time periodic signals, probably because a period has to both start and end at a sampling instant. Continuous-time periodic signals should be able to have any frequency. Indeed, give me any "segment" or snippet of continuous-time signal, of any finite time in length, and I can connect the end to the beginning (like in a "tape loop" or circular buffer) and voila, it is exactly periodic and extends infinitely in both directions.