# Are locally integrable functions almost completely determined by their approximate modulus of continuity?

This is a follow up to this question, which was answered in the affirmative: Are continuous functions almost completely determined by their modulus of continuity?

Note: We do not identify functions that agree a.e.

Given $$f: R \to R$$ in $$L^{1}_{loc}$$, define the approximate left modulus of continuity, $$L_f (x, e): \mathbb R \times \mathbb R^+ \to [0, \infty]$$ by

$$L_{f} (x, e)=\sup \{ d \geq 0\: \big|\, A(f, r) \leq e, \forall r \text{ s.t. } 0 \leq r \leq d \}.$$

where $$A(f, r) := \frac{1}{r} \int\limits_{[x-r, x]} |f(t) - f(x)| {d}t.$$

Similarly define the approximate right modulus of continuity by

$$R_{f} (x, e)=\sup \{ d \geq 0\: \big| I(f, r) \leq e, \forall r \text{ s.t } 0 \leq r \leq d \}.$$

where $$I(f, r) := \frac{1}{r} \int\limits_{[x, x + r]} |f(t) - f(x)| {d}t.$$

Note that if $$f = g$$ a.e., then $$L_f = L_g$$ a.e. and $$R_f = R_g$$ a.e., by which we mean for almost all $$x$$, for all $$e$$, $$L_f (x, e) = R_f (x, e).$$

Do $$L_f$$ and $$R_f$$ almost determine f uniquely? In the following sense:

Suppose $$f$$ and $$g$$ in $$L^{1}_{loc}$$ are such that $$L_f = L_g$$ a.e. and $$R_f = R_g$$ a.e. Does it follow that $$f = g + c$$ a.e. or $$f = -g + c$$ a.e. for some a.e. constant function $$c$$?