Can $\kappa$-club is defined for any class of ordinals?

Question

In reading "Proof Theory - The First Step into Impredicativity", I'm stuck at Thm. 3.2.19.

3.2.19 Theorem Let $\kappa$ be a regular ordinal. A class $M\subseteq On$ is $\kappa$-club iff $en_M$ is a $\kappa$-normal function.

And in "$\Leftarrow$" part of the proof,

Because of $\kappa\subseteq dom(en_M)$ we obtain $otyp(M\cap\kappa)=\kappa$.

My main question is "Why is this true?".
In my opinion, if $M\cap\kappa=\emptyset$, $\kappa\subseteq dom(en_M)$ may be true and $otyp(M\cap\kappa)=\kappa$ never. Moreover this situation is caused because $\kappa$-club is defined for any class of ordinals in this book. In order to be unbounded in $\kappa$, $M$ have to contain ordinals less than $\kappa$(, so $otyp(M\cap\kappa)=\kappa$). $M$ containing only ordinals over $\kappa$, however, may make it true that $en_M$ is a $\kappa$-normal function.

Answer 1

I don't have the book at hand, but if I recall correctly the following definition of normality is used:

A function $f$ is $\kappa$-normal if $(i)$ the domain of $f$ is $\kappa$, $(ii)$ the range of $f$ is contained in $\kappa$, $(iii)$ for $\alpha<\beta<\kappa$ we have $f(\alpha)<f(\beta)$, and $(iiv)$ for $\lambda<\kappa$ limit we have $f(\lambda)=\sup\{f(\eta): \eta<\lambda\}$.

If so, then the key point is bulletpoint $(ii)$: by restricting the range we prevent exactly the situation you're worried about. If, for example, the smallest element $\alpha$ of $M$ is greater than $\kappa$, we have $f(0)=\alpha>\kappa$ - and thus $f$ can't be $\kappa$-normal.