Properties of the Tangent Function

From the graph:

1. The curve is not continuous — it has (removable-in-appearance but genuine) discontinuities at every $x=(2n+1)\tfrac{\pi}2$, $n\in\mathbb{Z}$.
2. The branch on $\left(-\tfrac{\pi}2,\tfrac{\pi}2\right)$ is symmetric about the origin (odd).
3. There are infinitely many vertical asymptotes, $x=(2n+1)\tfrac{\pi}2$, $n\in\mathbb{Z}$.
4. Tangent has neither a maximum nor a minimum value — it runs unboundedly to $\pm\infty$ near every asymptote. …