I was reading through the proof of the existence of the Gibbs phenomenon for the ramp function $$ f(t) = \frac{\pi}{2} \mathrm{sgn}(t) - \frac{t}{2} $$ at $t=0$, where the author states the following without any explanation as to where the $\chi$-function comes from: $$ \ldots = \int_{0}^{\frac{\pi}{n}} \sin(nt)\left( \frac{ \cos\left( \frac{ t }{ 2 }\right) }{ 2\sin \left( \frac{t}{2} \right) } - \frac{ 1 }{ t } \right)\,\mathrm{d}t = \int_{0}^{\pi} \sin(nt)\left( \frac{ \cos\left( \frac{ t }{ 2 }\right) }{ 2\sin \left( \frac{t}{2} \right) } - \frac{ 1 }{ t } \right) \chi_{[0,x_n]}(t)\,\mathrm{d}t\,, $$ where $x_n = \frac{\pi}{n}$. The author then goes on to define the function $g$ as follows: $$ g = \left( \frac{ \cos\left( \frac{ t }{ 2 }\right) }{ 2\sin \left( \frac{t}{2} \right) } - \frac{ 1 }{ t } \right) \chi_{[0,x_n]}(t) $$ and develop a continuous extension $$ G_n = \left\{ \begin{aligned} &g_n(t) &&,\quad -\pi\leq t\leq \pi,\quad t\neq 0\\ &0 &&,\quad t=0 \end{aligned} \right. $$ for it in order to use the ML-theorem from complex analysis to prove the convergence of the integral.
My question is, what is the meaning of $\chi_{[0,x_n]}(t)$ in this context? Some Googling resulted in me discovering the Legendre chi function, but in its case the sub-index $\nu$ is usually a natural number, whereas here it seems to be an interval on the real line.
I'm baffled by this to say the least. I should say this is for my bachelor's thesis, so I would also appreciate any suggestions concerning literature on the subject.
The notation is often (as it is here) used to denote the indicator function, specifically $\chi_A(x)$ is set to $1$ if $x\in A,$ and $0$ otherwise.
In your case, the integration is effectively restricted to the interval $[0,x_n].$