Let $f$ be a locally integrable function on $\mathbb R^n$. By Lebesgue differentiation theorem, for almost every point $x$, one has $$ \lim_{m(B) \to 0} \frac{1}{m(B)} \int_B f(y) dx = f(x). $$ I want to know that, if we replace the $\frac{1}{m(B)} \int_B f(y) dx$ by $\operatorname{ess\,sup} f(y)$, is the statement remains true?
If it is true, is $\lim_{m(B) \to 0} \operatorname{ess\,sup} |f(y) - f(x)| = 0$ almost everywhere?
If not, what could we say about the limit of esssup function?