Clubs are ranges of normal enumerations

The inherited ordinal order enumerates $C$ by an ordinal $\eta \le \kappa$, successively taking the least unused point. Unboundedness and regularity force $\eta \ge \kappa$, hence $\eta =\kappa$. The least-unused rule also proves uniqueness.

If $C$ is closed and $0<\lambda <\kappa$ is limit, $\delta ={\sup }_{\xi <\lambda }e(\xi )<\kappa$ is a limit point of $C$, hence lies in $C$. Strict increase and least-unused enumeration force $e(\lambda )=\delta$. Thus $e$ is normal.

Conversely, if $e$ is normal and $\delta <\kappa$ is a nonzero limit point of $C$, the indices of the points in $C\cap \delta$ form an initial segment $\lambda <\kappa$ with no last element. (They cannot be all kappa since C is unbounded.) Continuity gives $e(\lambda )=\delta$, so $C$ is closed. Finally, a strictly increasing map on kappa has unbounded range: a bounded range cannot contain kappa distinct ordinals. It is the increasing enumeration of its range, giving the last equivalence.