Is it possible to choose the Radon-Nikodym derivative to be continuous almost everywhere?

Question

I am considering a compact Hausdorff topological space $X$, and the space of complex Borel Radon measures on $X$. Given a measure $\mu$ in this space, the polar decomposition allows us to write

$$\mu = e^{i\theta}|\mu|$$

for some Borel integrable (wrt $|\mu|$) function $e^{i\theta}$. Is it possible to choose this function to be continuous almost everywhere? (If not, when is it possible?)

In general is this possible for measures?

Thanks!