Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 105: 3

Discontinuous at $k\pi$, where $k$ is an integer.

Note that if $f(x)$ is continuous for an interval, then $|f(x)|$ is also continuous as every point and is reflected across the x-axis if negative and remains the same if positive. Thus, the only discontinuities of $|cot(x)|$ are the discontinuities of $cot(x)$ which occur at $k\pi$, where $k$ is an integer.