First let's have some definition (you can find these in any complexity's textbook)
Theorem. [Gap Theorem] There is a computable time bound $t(n)$ such that DTIME[ $t(n)$ ]= DTIME[ $r(t(n))$ ] for any computable function $r$.
This theorem says that even if you have more time resources, then you cannot do more. We have another theorem that says.
Theorem. [Time Hierarchy Theorem (THT)] if f and g are time-constructible functions and f(n)=o(g(n)), then DTIME[ $f(n)$ ] $\subsetneq$ DTIME[ $g(n)$ ].
It is clear that if $f$ and $g$ are not time-constructible, then THT wouldn't hold. Time-constructible ensures that we read the whole input in the input tape of the TM.
Question: If $f$ and $g$ are not time-constructible, why then THT cannot hold? What makes THT cannot hold even if f and g are not time-constructible?