In *How to recognize constant functions. Connections with Sobolev spaces* (Russian Math Surveys **57** (2002), H. Brezis recalls the following fact:
> Let $\Omega\subset{\mathbb R}^N$ be connected and $f:\Omega\rightarrow{\mathbb R}$ be measurable, such that 
$$\int\int_{\Omega\times\Omega}\frac{|f(y)-f(x)|}{|y-x|^{N+1}}\,dx\,dy<\infty.$$
Then $f$ is constant.

He adds 
> *The conclusion is easy to state, but I do not know a direct,elementary, proof. Our proof is not very complicated but requires an “excursion” via the Sobolev spaces.*

My question is whether there is such an elementary proof in the special case of one space dimension ($N=1$, $\Omega$ an interval).