non time constructible functions 
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?
 A: 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.
A: According to the definition of Arora, trigonometric functions are not good counter-examples since they are not functions from $\mathbb{N}$ to $\mathbb{N}$.
However, the function $TUC : \mathbb{N} \rightarrow \mathbb{N}$ defined by $TUC(n) = n$ if $M_n(n) \neq n$ and $TUC(n) = 2n + 1$ otherwise, is not time-constructible:
Suppose there exists $M$ that computes $TUC$ in $TUC(n)$ time, then there exist $n$ such that $M = M_n$ (cf. encoding of a Turing machine in Arora's book), but 
-if $TUC(n) = n$, then $M(n) \neq n$ and so $TUC(n) \neq n$ $\rightarrow$ contradiction.
-if $TUC(n) = 2n +1 $, then $M(n) = n$ and so $TUC(n) = n$ $\rightarrow$ contradiction.
Both cases end up to a contradiction so $TUC$ is not time-constructible.
