Is the product of two non-holomorphic function always non-holomorphic?
Yes, consider $\bar z \cdot \dfrac 1{\bar z} = 1$. However, if $f$ is holomorphic and $h$ is holomorphic, then $g=h/f$ is holomorphic wherever $f\ne 0$.
Yes, consider $\bar z \cdot \dfrac 1{\bar z} = 1$. However, if $f$ is holomorphic and $h$ is holomorphic, then $g=h/f$ is holomorphic wherever $f\ne 0$.