Fundamental period of the function $f(x) = \left\{\begin{matrix}
2\;\;\;\;,x\in \mathbb{Q} \\\\
-2\;\;\;\;,x\notin \mathbb{Q} &
\end{matrix}\right.$ is
As we know that function $f(x)$ is periodic function, If it satisfy the relation $f(x+T)=f(x)$
and smallest positive value of $T$ is called period of that function $f(x)$.
But here i did not understand How can i calculate period of that constant function
Help me,Thanks
Best Answer
This function has no fundamental period. Suppose that $T$ were such a period. Then it would have to be rational, for $f(0) = F(T) = 2$. Consider $T' = T/2$. This, too, will be a period, for if $x$ is rational, so is $x + T'$, and if $x$ is irrational, so is $x + T'$.
In fact, the numbers $T = 1, 1/2, 1/4, \ldots$ are all periods of this function, so there's no smallest positive period.