Affine rigidity of functions with additive oscillation

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 $\mathrm{\mid }Df\mathrm{\mid }(I)=4\text{osc}(f,I)$ for all intervals $I\subset (a,b)$ must be affine.