Question:
The function f (x) = tan x is discontinuous on the set
(a) $\{n \pi: n \in Z\}$
(b) $\{2 n \pi: n \in Z\}$
(c) $\left\{(2 n+1) \frac{\pi}{2}: n \in Z\right\}$
(d) $\left\{\frac{n \pi}{2}: n \in Z\right\}$
Solution:
(c) $\left\{(2 n+1) \frac{\pi}{2}: n \in Z\right\}$
When $\tan (2 n+1) \frac{\pi}{2}=\tan \left(n \pi+\frac{\pi}{2}\right)=-\cot (n \pi)$, it is not defined at the integral points. $[n \in Z]$
Hence, $f(x)$ is discontinuous on the set $\left\{(2 n+1) \frac{\pi}{2}: n \in Z\right\}$.