Chapter 1 - Limits and Continuity - 1.6 Continuity of Trigonometric Functions - Exercises Set 1.6 - Page 105: 2

Discontinuous at $\pi$

Note that $cos(x)$ is continuous for all values of $x$. Thus, the only discontinuities for $f(x) = cos(\frac{x}{x-\pi})$ is when $\frac{x}{x-\pi}$ is not a real number. This occurs when the denominator, $x-\pi$, equals $0$, which evaluates to $x=\pi$.