I have the definition of a periodic function in my Real Analysis "textbook" given by my professor and below is a remark that states "A function can be periodic without its period being defined"
The definition given is : Let E ⊂ R and t ∈ R*+ (positive real numbers). A function of f : E → R is t-periodic if, for all x ∈ E, we have x + t ∈ E and f(x + t) = f(x). The smallest t > 0 with this property, if it exists, is called the period of f.
What does it mean for the period to be undefined? And what are some examples of periodic functions with an undefined period? (if the examples could be a bit elaborated that would really really help)
Thank you so much in advance and if you have any good sources for Real Analysis to help with the understanding of concepts that would be amazing. I'm taking my first real analysis course and having issues wrapping my head around certain definitions because I'm a visual learner and it's hard for me to learn from definitions.
Based on the standard definitions for periodic functions and period, a function $f:\mathbb{R}\to\mathbb{R}$ is said to be periodic if for some $p>0$, for every $x\in\mathbb{R}$, $$f(x+p) = f(x).$$ Among such $p>0$, if there is a smallest one, then we call the smallest $p$ as the period of $f$. But there can be no smallest such number. For example, let $f(x) = 0$. Then for any $p>0$ we have $f(x+p) = 0 = f(x)$. More complicated example will be the Dirichlet function, $$\chi_{\mathbb{Q}}(x) = \begin{cases} 1 &x\in\mathbb{Q}\\ 0 &x\notin\mathbb{Q}\end{cases}.$$ This is periodic since for any positive rational $p$, $f(x+p)=f(x)$(take case when $x$ is rational or irrational). So there's no smallest such $p$, and its period is not defined. This is the case your professor is talking about.