Is a constant function periodic? Can we regard a constant function "$f(x)=\text{constant}$" to be a periodic function? If yes, what is its period?
 A: Yes, a constant function is a periodic function with any T∈R as its period (as f(x)=f(x+T) always for howsoever small 'T' you can find).
However, the fundamental period of a constant function is not defined for the above reason.
A: Nowhere in the definition of a period function is it stated that the function must have a least period.
If $f(x) = c$ then for any $p$ we have $f(x+p) = f(x)$.  So $f$ is periodic and $p$ is a period.  Obviously any other non-negative value will also be a period.
There is nothing in the definition of periodic function that says that is not allowed.
A: 
Definition 1: For $P>0$, a function $f:\Bbb R\to\Bbb R$ is periodic of period $P$ (or $P$-periodic) if $$\forall x\in\Bbb R,\ f(x+P)=f(x)$$ A function is commonly said to be periodic if there is some $P>0$ such that $f$ is $P$-periodic.
Definition 2: For a periodic function $f$, the fundamental period of $f$ is, if existing, the least positive real number $T$ such that $f$ is $T$-periodic. Specifically, if we call $$L=\inf\{ P>0\,:\,\forall x\in\Bbb R,\ f(x+P)=f(x)\},$$ then $f$ has a fundamental period if and only if $L>0$; in which case, $L$ is the fundamental period of $f$.

Answer to your question: Constants are periodic functions of any period and, therefore, they do not have a fundamental period.
