Consider Example 3.5 in the following lecture notes on Fourier Analysis, on page 10 at the bottom.
http://www.math.ku.dk/~schlicht/DL/2013/fourier-summary.pdf
I cannot understand why it says that a function defined on $[-\pi,\pi[$ is $2\pi$-periodic. It further says that $f$ is discontinuous for all points $n\pi$ where $n$ is an integer, but it is not even defined in $\pi$, let alone $n\pi$?
Could someone help me understand what's going on here?
He's not saying that a function defined on $[-\pi,\pi[$ is $2\pi$-periodic. He's defining a $2\pi$-periodic function by specifying its values on $[-\pi,\pi[$, and then extending it periodically to all of $\Bbb{R}$.
I think the problem is that the notes are unfortunately worded. They: "Let $f : \Bbb{R} \to \Bbb{R}$ be the $2π$-periodic function on $[−π, π[$ given by (stuff)." It would be better to phrase that as "Let $f : \Bbb{R} \to \Bbb{R}$ be the $2π$-periodic function given on $[−π, π[$ by (stuff)." There seem to be a few other word-order issues like this in the notes; I suspect that they were not written by a native English speaker.
