Periodic function without trigonometry and complex numbers

Question

Can I get a periodic function without using trigonometric functions or complex numbers?

UPDATE:

The question has been superseded by https://math.stackexchange.com/questions/1415757/single-statement-continuous-periodic-function-without-trigonometry-and-complex-n

Answer 1

This function has period 2. It equals $\lfloor x\rfloor\pmod2$ $$f(x)=\left\{\begin{array}{l}0,\,\, 2n\leq x<2n+1\\1,\,\, 2n+1\leq x<2n+2\end{array}\right.$$

Answer 2

This question is badly defined when you think about it. You can specify a periodic function simply by giving the function on the period and specifying it's periodically continued. Nobody says every function has to be given in the way you are used to from school (a sum or product of existing functions). So, it's perfectly reasonable to turn any function $$g(x),0\leq x <1$$ into a periodic function $$f(x)=g(x-\lfloor x \rfloor)$$ where you have the fractional part in the parentheses.

On the other hand, periodic function can (usually) be expressed with Fourier series, so every periodic function can also be given as an infinite sum of sines and cosines. Most functions you see in handbooks with given fourier series are piecewise periodically extended functions (triangle wave, square wave,...).

So the answer is... any function can be periodically extended... and also expressed with trigs if you want.