Affine rigidity of functions with additive oscillation

Abstract

We prove that a locally integrable function $f:(a,b) \to \mathbb R$ must be affine if its mean oscillation, considered as a function of intervals, can be extended to a locally finite Borel measure. In particular, we show that any function $f$ satisfying the integro-differential identity $|Df|(I)=4\text{osc}(f,I)$ for all intervals $I \subset {(a,b)}$ must be affine.