Can a function in $\mathcal{W}^{1,2}((a,b); \mathbb{R})$ be unbounded?

Question

I want to see an example of unbounded functions in the space $\mathcal{W}^{1,2}((a,b); \mathbb{R})$.

Answer 1

As you may know, $W^{1,2}(a,b)$ can be embedded into $C[a,b]$. (c.f. https://en.wikipedia.org/wiki/Sobolev_inequality). So every function in $W^{1,2}(a,b)$ is bounded.