Points at which a function is holomorphic $f=\frac{1}{z^5-1}$
$z=1$ ofcourse makes it non-holomorphic.
What other z's would make it non-holomorphic? 
Is it only $z=1$ ?

$z^5-1=0$ has $5$ roots, the fifth roots of 1:
$$
z=e^{\tfrac{k\,\pi}{5}\,i},\quad 0\le k<5.
$$ Only one of them is real.
