What will be the Turing machine to show that the function $T(n) = n$ is time constructible?
You almost certainly have the right machine: just implement a binary counter and increment it once for each character of the input.

What you need is a more sophisticated analysis of the running time. You've presumably reasoned that the counter will end up with $\log n$ bits, you need to increment it $n$&nbsp;times and each increment could change all the bits, so there are $O(n\log n)$ operations.  This is completely correct but you can do better.

Suppose the current value of the counter is even. How many bits do you have to change to increment the counter? How often do you need to change more than two bits? How often do you have to change all $\log n$ bits to increment the counter?