Non constructible functions and anomalous results In the Arora-Barak book, in the definition of time constructible functions, it is said that using functions that are not time constructible can lead to "anomalous results". Does anyone have an example of such an "anomalous result"? I have heard in particular that there might exist functions such that the time hierarchy theorem does not hold, does anyone have an example of such functions? Is there something about this somewhere in the litterature?
Borodin's Gap Theorem: For every total computable function $g(n) \geq n$, there is a total computable function $t$ such that $DTIME[g(t(n))] = DTIME[t(n)]$. 

In fact, this holds for any Blum complexity measure in place of $DTIME$. 

See also the wikipedia page and references therein.