Graph of the Cosecant Function

On $(0,2\pi)$, cosecant is continuous everywhere EXCEPT at $x=\pi$ (where $\sin\pi=0$), and it has neither a maximum nor a minimum. Its behaviour, piece by piece: falls from $+\infty$ down to $1$ as $x$ runs over $\left(0,\tfrac{\pi}2\right]$; rises from $1$ back up to $+\infty$ as $x$ runs over $\left[\tfrac{\pi}2,\pi\right)$; then, on the other side of the asymptote at $\pi$, rises from $-\infty$ up to $-1$ as $x$ runs over $\left(\pi,\tfrac{3\pi}2\right]$; and falls from $-1$ back down to $-\infty$ as $x$ runs over $\left[\tfrac{3\pi}2,2\pi\right)$. …