Continuity and other properties of complex exponential

Question

So I think I can do the others, but part (i) about showing the continuity of $a^z$ has me stumped. I always get really stuck when it comes to proving continuity (I am using the metric spaces definition.)

Answer 1

$P(z)= a^z = E(A(z))$, where $E(z)=\exp(z)$ and $A(z)=z \log a$.

Both $E$ and $A$ are continuous functions and so is $P = E\circ A$, being a composition of continuous functions.

This argument assumes you have proved that $\exp$ is continuous, which should come from its power series definition. The function $A$ is simply a scaling.