A function $T: \mathbb N \rightarrow \mathbb N$ is 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 $T(n)$. ($\llcorner T(\vert x\vert) \lrcorner$ denotes the binary representation of the number $T(\vert x\vert)$.)
Examples for time-constructible functions are $n$, $nlogn$, $n^2$, $2^n$. Time bounds that are not time constructible can lead to anomalous results.
--Arora, Barak.
This is the definition of time-constructible functions in Computational Complexity - A Modern Approach by Sanjeev Arora and Boaz Barak.
It is hard to find valid examples of non-time-constructible functions. $f(n)=c$ is an example of a non-time-constructible function. What more (sophisticated) examples are out there?
You can use the time hierarchy theorem to create a counter-example.
You know by the THT that DTIME($n$)$\subsetneq$DTIME($n^2$), so pick a language which is in DTIME($n^2$)$\backslash$DTIME($n$). For this language you have a Turing machine $A$ which decideds if a string is in the language in time $O(n^2)$ and $\omega(n)$. You can now define the following function
$f(n) = 2\cdot n+A(n)$
If it is time constructible it contradicts the THT.