Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 105: 8

Discontinuities at $x = \frac{\pi}{2} + k \pi$, where $k$ is an integer.

Discontinuities exist in $f(x) = \sqrt{2+tan^2(x)}$ where $tan(x)$ is undefined or where $2+tan^2(x)<0$. $tan(x)$ is undefined at $x = \frac{\pi}{2} + k \pi$, where $k$ is an integer. Solving for $x$ where $2+tan^2(x)<0$, we find $$2+tan^2(x)<0$$ $$x = \frac{\pi}{2} + k \pi$$