I'm working with the following definition:
$Def1$: We say that a number $τ ∈ ℝ$ is a period for the function $f : ℝ → ℝ$ if $x ∈ ℝ ⇒ f(x+τ) = f(x)$. A function with a positive period is said to be periodic. If $f$ is periodic and exists $τ_0 = min\{τ > 0 |x ∈ ℝ ⇒ f(x+τ) = f(x)\}$, then we say that $τ_0$ is the fundamental period of $f$.
Using this definition I was able to prove that:
$Th1$: If $f : ℝ → ℝ$ has period $P > 0$ and is continuous and non-constant on $[0,P)$, then $f$ has a fundamental period.
Now $Def1$, requiring for the function to be definite on $ℝ$, seems too restrictive to me, after all there are plenty of functions which we call periodic and don't comply with this requirement (e.g. tan). I was thinking of switching to something like:
$Def2$: We say that a number $τ ∈ ℝ$ is a period for the function $f : A ⊆ ℝ → ℝ$ if $x ∈ ℝ ⇒ x±τ ∈ A ∧ f(x+τ) = f(x)$. A function with a positive period is said to be periodic. If $f$ is periodic and exists $τ_0 = min\{τ > 0 | x ∈ A ⇒ [x±τ ∈ A ∧ f(x+τ) = f(x)]\}$, then we say that $τ_0$ is the fundamental period of $f$.
My question is:
Is $Def2$ a correct formulation of the notion of periodicity, and does the analogous of $Th1$ still holds for $Def2$?
Proving that, if $τ_0 = inf\{τ > 0 | x ∈ A ⇒ [x±τ ∈ A ∧ f(x+τ) = f(x)]\}$, then $x ∈ A ⇒ x+ τ_0 ∈ A$ doesn't seem trivial.
There's not really a definite answer your first question, since "a correct formulation of the notion of periodicity" is quite subjective. It's certainly totally reasonable to call such functions "periodic", though.
The analogue of $Th1$ still holds, and the proof is virtually identical. As a sketch, note that the set of periods of $f$ is a subgroup of $\mathbb{R}$, and so it is either discrete or dense in $\mathbb{R}$. If it is discrete and nontrivial there is a minimal period, and if it is dense in $\mathbb{R}$ then continuity of $f$ immediately implies $f$ must be constant.