Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function.

Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous?

Note: We use the “strong” definition of Lebesgue point, given here.

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function.

Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous almost everywhere?

Note: We use the “strong” definition of Lebesgue point, given here.

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function.

Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous?

Note: Here we use the “strong” definition of Lebesgue point, given here.

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function.

Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous?

Note: We use the “strong” definition of Lebesgue point, given here.