What is continuous in a domain up to boundary?

Question

The definition roughly states:-

A function $f(z)$ is said to be continuous in a domain D up to boundary if for each $a$ belong to domain or boundary we have $f(z) \rightarrow f(a) $ as $ \rho _D(z,a)\rightarrow 0$

Here the f is complex function

$\rho _D(x,y) $ is defined infimum of arc length of a curve connecting point $x$ and $y$ within $D$

Domain is an open connected subset of extended complex plane

The book says roughly that if the boundary point $a$ is not a self intersection point on the boundary curve or if a is an interior point of $D$ then the above limit equals limit $ f(z)$ as $z\rightarrow a$ , $z $ belongs to $D$

My question is :-

How to prove or disprove that the two limits are same even for when boundary point is a self intersection point, or if they are not same, why are the limits not same, what would be counter example?

Answer 1

How about an example like this. Let $S$ be a spiral in the plane approaching zero, and let $D = \mathbb C \setminus (S \cup \{0\})$. Suppose the spiral approaches $0$ so slowly that $\rho_D(z,0)=\infty$ for any $z \in D$. So it seems that "continuous up to boundary" does not require $f$ to be continuous at $0$.