Complex analysis is different #2

Let $A$ be the circumference $A = \{ z \in \mathbb{C}; \space |z| = 1\}$ and $B$ be the set $B = A - \{ -1 \}$ .

True or false?

a) There exists at least one continuous function $f : A \rightarrow \mathbb{C}$ such that given an $a \in A$ , $f(a)$ is one and only one of the roots of the equation $e^z = a$ .

b) There exist infinitum continuous function $f : B \rightarrow \mathbb{C}$ such that given an $a \in B$ , $f(a)$ is one and only one of the roots of the equation $e^z = a$ .