Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical point means $\kappa$ is the least ordinal such that $\kappa<j(\kappa)$; Note $j$ must take ordinals to ordinals and for all ordinals $\alpha\leq j(\alpha)$).
It is common knowledge (reference: Set Theory, by Jech pg. 289) that $j(V)=M$, i.e. $j$ is surjective. I must be misunderstanding something because I don't see how this can be possible when there are gaps between ordinals in $j(V)$?: If $\beta$ is an ordinal such that $\kappa<j(\beta)<j(\kappa)$ (for instance $j(\beta)=\kappa+1$) then $j(\beta)<j(\kappa)$ implies $\beta<\kappa$ hence $j(\beta)=\beta<\kappa$, which contradicts $\kappa<j(\beta)$.
Thanks.