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take [itex]f(z)=\ln{z}=\ln{r}+i arg(z)[/itex],

here [itex]z=z_0+2k \pi[/itex] all correspond to the same value of z but give different values of f(z) i.e. its a multifunction.

how does this affect its holomorphicity?

as far as i can tell, it doesn't and this function is holomorphic everywhere in [itex]\mathbb{C}[/itex]. Is this true?

Or is it the case that it isn't holomorphic on the negative real axis since (just as for f(z)=z), if we make a branch cut and choose only to work with the principal argument then we have a discontinuity in the principal argument on the negative real axis. Why does this discontinuity affect holomorphicity though?

Are there any useful things to note about multifunctions? All my notes give is a definition. I was just wondering if there was any properties I should know about for my exam?