In the book Introduction to Set Theory' by Hrbacek and Jech, chapter $6$ Ordinal Numbers, section $6$ Normal Form, I don't understand the definition of continuity of ordinal numbers.
Ordinal functions $\alpha+ \beta, \alpha \cdot \beta, \alpha^\beta$ are continuous in the second variable:
If $\gamma$ is a limit ordinal and $\beta=\sup_{\nu < \gamma} \beta_{\nu} $, then
$\alpha+\beta=\sup_{\nu < \gamma}(\alpha + \beta), \alpha \cdot \beta=\sup_{\nu < \gamma}(\alpha \cdot \beta_{\nu}), \alpha^{\beta}=\sup_{\nu < \gamma}\alpha^{\beta_\nu}$
Can anyone explain to me what is the meaning of continuity of ordinal functions?
Continuity just means that taking $\sup$ before or after applying the function gives the same result. This is the same with real numbers, that given a convergent sequence $x_n\to x$, then $f(x_n)\to f(x)$.
So when we say that ordinal addition is continuous we say that $$\sup\{\beta+\gamma\mid\gamma<\delta\}=\beta+\sup\{\gamma\mid\gamma<\delta\}=\beta+\delta.$$