So, given an infinite cardinal $\kappa$, what can we say about the continuity of the function $$\begin{align}f : \text{Card} &\longrightarrow \text{Card} \\ \lambda &\longmapsto \lambda^\kappa \end{align}$$ ? Can we characterize the class of limit cardinals which are discontinuity points? If it is a trivial matter, could you give me nonetheless a hint? Can we at least characterize a class of infinite cardinals that are continuity points?
Thanks