Working on a problem in Gamelin's book.
"Show that if $f(z)$ is a continuous function on a domain $D$ such that $f(z)^N$ is analytic on $D$ for some integer $N$, then $f(z)$ is analytic on $D$."
He starts a hint with the statement "Show that the zeros of $f(z)$ are isolated." This makes me think. I know that if $f$ is analytic and nonconstant, then the zeros are isolated. Is there some sort of converse to this theorem? I.e., for example, if continuous and zeros are isolated then analytic?
His next hint is: "At a zero of $f(z)$, write $f(z)^N=(z-z_0)^mh(z)$ where $h(z_0)\ne 0$ and show that $N$ divides $m$." However, the book has covered nothing on winding numbers thus far. Also, how will the fact that $N$ divides $m$ help to show that $f(z)$ is analytic?