**Definitions and some motivation:**

Let $\mathcal B$ be the set of bounded measurable functions from $[0, 1]$ to $\mathbb R$. Denote by $\mathcal N$ the set of measurable subsets of $[0, 1]$ with Lebesgue measure $0$.

Given a function $f \in \mathcal B$, define the function $\mathcal Of$ by

$\mathcal Of(x) := \inf_{N \in \mathcal N} \lim_{\delta \to 0} \sup_{y, z \in B_\delta (x) \setminus N}\ |f(y) - f(z)|$.

Thanks to Lusin’s theorem, we know that we can modify $f$ on an arbitrarily small set and get a continuous function, and so we force the oscillation to be $0$ everywhere. But can we force it to be whatever we want, and in a controlled way?

**Question:**

Does there exist, for any given $f, g \in \mathcal B$ and $\varepsilon > 0$, a function $f’ \in \mathcal B$ such that the following conditions are satisfied?

i) $f’ = f$ everywhere except for a set of measure at most $\varepsilon$.

ii) $\mathcal Of’ = \mathcal Og$.

iii) $\mathcal O(f’ - f) \leq |\mathcal O(g) - \mathcal O(f)|$.

**Remark:** Condition (iii) intuitively says that we do not modify the values of $f$ any more than strictly necessary.

*Note: All functions are genuine functions and not equivalence classes modulo null sets of such.*