Periodic functions

Concentrate on the function represented in the previous figure. The images of $$\ldots,-4,-2, 0, 2, 4,\ldots$$ coincide and they are equal to $0$.

In fact we can observe that the image of any real number $x$ with the images of $$x + 2, x + 4,\ldots$$ We will say that the function is periodic.

A function $f$ is periodic of period $T$ if it exists a positive real number $T$ such that for any $x$ of the domain we have that:$$f (x + T) = f (x)$$Note that if $T$ is a period of the function then also any multiple of $T$ is also a period. The minimum value of $T$ that satisfies the previous definition is known as the fundamental period.

Find the fundamental period of the following function:

We observe that the function takes the value $0$ in each integer, and has the same behaviour between any $n$ and $n + 1$.

Therefore, since there are no other regularity for shorter periods, we will have that the fundamental period of $f$ is $T = 1$.