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## Homework Statement

Let f:[a,b] -> R be a bounded function, and let D be the set of points at which f is not continuous.

(a) Prove that D is a countable union of closed sets.

(b) Prove that if m(D) = 0, then f is measurable.

## Homework Equations

Of(x) = lim(ε->0)(sup f(y) - inf f(y) ) on {y:|y-x|< ε} [Called the "Oscillation at x"]

## The Attempt at a Solution

I know that if m(D) = 0, then f is Reimann integrable. This probably leads to a fact about Lebesgue integrability, which leads to a fact about measurability...? I'm not sure how it all ties together, or what the formal argument would be.

Any help would be much appreciated : )