I am self studying complex analysis and I have a question about holomorphic functions: Are singularities the only thing that makes a function nonholomorphic ? in other words, is there a complex function that is defined for example in the whole complex plane (i.e. no singularities ) which is holomorphic on some region but nonholomorphic in another region.
I think such function does not exist because the values of a holomorphic function in different places of the complex plane are linked by the Cauchy integral formula, but I am not sure about this.
$f(z)=\overline{z}$ doesn't have any singularities and is not holomorphic.
We can achieve the function being holomorphic on a region with a piece-wise definition. Let $f(z)=z$ on the unit disc and $f(z)=\overline{z}^{-1}$ everywhere else. Not only does this satisfy your desire, it's also continuous, since $\overline{z}=z^{-1}$ holds on the unit circle.