I know that f(n) = n is a computable, weakly time constructible function but NOT a time constructible function.
But I can't think of any computable function that is not weakly time constructible.
Can anybody help with it?
A function $T: \mathbb N \rightarrow \mathbb N$ is weakly time constructible if $T(n) \geq n$ and there is a $TM$ $M$ that computes the function $x \mapsto \llcorner T(\vert x\vert) \lrcorner$ in time $O(T(n))$.
( $\llcorner T(\vert x\vert) \lrcorner$ denotes the binary representation of the number $T(\vert x\vert)$.)