We define say that a function $f\in L^1_{\mathrm{loc}}(\mathbb R)$ satisfies property (X) if every point $x\in \mathbb R$ is a Lebesgue point, i.e. $$\lim_{\varepsilon\to 0}\frac 1{2\varepsilon}\int_{x-\varepsilon}^{x+\varepsilon}f(y)\,\mathrm dy=f(x)$$ for all $x\in\mathbb R$. Clearly all continuous functions satisfy (X), as do certain discontinuous functions like $\mathbf1_{\{x\ne0\}}\sin(1/x)$ or $(\operatorname{sgn} x)|x|^{-\alpha}$ for $0<\alpha<1$. But are there any positive results we can give about properties implied by property (X)?
It would be particularly useful for me to find properties implied by property (X) together with nonnegativity.
I am unable to post this as a comment, so I put it as an answer instead, although it is not. Your $f$ is a Baire class 1 function, which I guess is a trivial implication of (X). Regarding the set of discontinuity, see for example https://mathoverflow.net/q/32033/32658. Now, if $f$ is a Baire-1, locally summable function, let $\{f_n\}$ be a sequence of continuous functions converging to it pointwise; we have $$|f(x)-\frac{1}{2\epsilon}\int_{B(x,\epsilon)} f(y)\;dy|\leq |f(x)-f_n(x)|+|f_n(x)-\frac{1}{2\epsilon}\int_{B(x,\epsilon)} f_n(y)\;dy|+\frac{1}{2\epsilon}\int_{B(x,\epsilon)} |f_n(y)-f(y)|\;dy$$ from which you could get some (again trivial) sufficient conditions for a Baire-1 function to satisfy (X). These are obvious observations, as David mentioned, if I didn't miss something.