In my first complex analysis for physicists course I was introduced to monodromy theorem. The statement of this theorem should be the following:
Given an open disk $U$ on the complex plane centered at a point $P$, a holomorphic function $f : U → C$ and another point Q in the complex plane, if $\gamma_0$ and $\gamma_1$ are homotopic and and for each $s ∈ [ 0 , 1 ]$ it is possible to do an analytic continuation of $f$ along $γ_s$ then the analytic continuations of $f$ along $γ_0$ and $γ_1$ will yield the same values at Q .
When we consider the restriction of the principal branch of $logz$ or $\sqrt z$ on some disk the theorem cannot be applied, otherwise we could define an analyitic continuation of the two function on the whole complex plane, which I think it is not possible because we must always exclude some sort of curve.
My question is: exactly which one of the conditions imposed by the monodromy theorem is not respected by the functions listed above ?