Branch points of a function $f(z)=g(z)h(z)$

Question

Let us say we have a function, $f(z)=g(z)h(z)$ where we know that $h(z)$ is a single valued. Can we say that $z=z_0$ is a branch point of $f(z)$ iff it is also a branch point of $g(z)$? If this is true does it still hold if some of the functions are not analytic at that point?