Answer
The function is continuous.
Work Step by Step
We know that $\tan x $ is continuous on the interval $(-\frac{\pi}{2},\frac{\pi}{2})$ and $\sin x $ is continuous everywhere. Moreover, the range of $\sin x $ is $[-1,1]\subset (-\frac{\pi}{2},\frac{\pi}{2})$. Then, by Theorem 5 of the composition functions, the function $$ f(x)=\tan (\sin x) $$
is continuous.
