Why is the map analytic?

Question

The map $$z\mapsto \frac{z-i}{z+i}$$ is an analytic isomorphism of the upper half-plane $\mathbb{H}$ and the unit disc $D=\lbrace w \in \mathbb{C}||w|<1 \rbrace$ .

So , what I do not undertand is that since $-i \ne \mathbb{H}$ the map is analytic . Analytic means holomorphic .Then I know the function $\frac{z-i}{z+i} $ is continuous , but continuous does not imply holomorphic .

Answer 1

It is an analytic map because it is the quotient of two analytic maps.