Measurability of a function f which is discontinuous only on a set of measure 0.

[glacier302]

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 : )

 

[Eynstone]

(a) Let M be the upper bound for f & consider the sets
O_n={x in D |2M/(n+1)<=Of(x)<=2M/n} n=1,2,..
Do you see why each O_n must be closed & their union is D?

(b)If D is a null set,f would be riemann integrable(& hence measurable) as it is bounded .