# Function with two periods

by Office_Shredder
Tags: function, periods
 Emeritus Sci Advisor PF Gold P: 4,500 Suppose a function f has two periods a, b with a =/= n*b or vice versa. Then is f necessarily a constant? What if a is not a rational multiple of b? Answers involving whether f is continuous or not are appreciated. Basically, here's now I started: WLOG, suppose b > a. Then b-a < b, and either a < b-a or a > b-a. Either way, we've found a third period which does not satisfy the integral (or rational) multiplicity requirement. Since one of these periods is smaller than the other, we can subtract the smaller from the larger to get another smaller period. This process should be repeatable infinite times to get infinite discrete periods which tend to zero. But what is this enough for? EX: Suppose p is Pi and e is, well, e. Then p-e is a period, and e-(p-e) = 2e-p is a period. 2e-p - (p-e) = 3e-2p is a period, and so is 3e -2p + (2e-p) 5e-3p. So you can make a whole bunch of periods.... I don't know which of the periods I have listed here is the smallest (I'm too lazy to get a calculator out and do this for more than a couple runs), but I think you can see the process of shrinking the period. So you'll get periods of the form ki*a - mi*b, where, I believe, ki/mi approaches b/a. Thoughts?
 Sci Advisor HW Helper P: 2,020
 HW Helper P: 3,348 Ahh Im not too sure about anything, but i was wondering how a function could have to periods, and then I realised the most obvious answer of extending in more diemensions :)
 Sci Advisor P: 2,751 Function with two periods Actually I think the OP was referring to one dimentional functions something like y=sin(x) + sin(Pi x) for example. Rather than "having two period" (as far as I can see) such a function as aperiodic.
 Sci Advisor HW Helper P: 2,537 There are also fractal functions. For example, consider the characteristic function of the rationals (f(x)=1 if x is rational, and 0 otherwise) which has a periods equal to any rational number. Or, if you prefer non-rational ratios, you could go with the characteristic function of the algebraic numbers (restricted to the real line).
Emeritus