Regarding the behaviour of conformal maps along the boundary, we had a definition of an analytic curve. I just can't wrap my head around this definition. It goes as follows:
A connected curve $\Gamma \subset \hat{\mathbb{C}}$ is called analytic, if there is an injective parametrisation $\gamma\colon[a,b)\to \Gamma$ which, for every $t\in(a,b)$, has an injective holomorphic continuation in a neighbourhood $B_δ(t)$ with $\delta=\delta(t)>0$.