I'm reading Sipser's Introduction to the Theory of Computation, and I'm reading about space-constructible functions. He gives the following definition:
A function $f: \mathbb{N} \to \mathbb{N}$ is space constructible if the function that maps the string $1^n$ to the binary representation of $f(n)$ is computable in space $O(f(n))$.
He follows up by saying,
All commonly occurring functions that are at least $O(\log n)$ are space constructible, including the functions $\log_2 n$, $n \log_2 n$, and $n^2$.
My question: What is an example of a function that is $\Omega(\log n)$ that is not space constructible?