Periodicity of a constant signal!
As you say, the constant function is periodic. A signal $x(t)$ is said to be periodic with period $p$ or to have a period $p$ if there exists a $p&gt;0$ such that $x(t+p)=x(t)$ for all real numbers $t$. Note that I said "a period" instead of "the period" -- periodic signals have an infinite number of periods, maybe even an uncountable infinite of them!

Often, what is referred to as the period of a signal is actually more formally described as the "fundamental period", which is the smallest among all periods.

So, a constant signal is periodic, it has an uncountably infinite number of periods (since any real number $p&gt;0$ is a period), but it does not have a fundamental period.