Suppose that f has a measurable domain and is continuous except at a finite number of points. Is f neccessarily measurable?Let A be the set of points where f is not continuous. Now since A is finite we have that m ( A ) = 0. Now as continuous maps are measurable if E is the measurable domain, then f defined on E ∖ A is measurable.Now I have a theorem that statesFor a measurable subset D of E, f is measurable on E if and only if f ∣ D and f ∣ E ∖ D are both measurable. f ∣ E ∖ A being measurable follows from the fact that f is continuous on E∖ A, but how can I show that f is measurable on A? I only know that the measure of A is zero, but nothing about the behavior of f there?

Question

Suppose that   f has a measurable domain and is continuous except at a finite number of points. Is   f neccessarily measurable?Let   A be the set of points where   f is not continuous. Now since   A is finite we have that   m  (  A  )  =  0. Now as continuous maps are measurable if   E is the measurable domain, then   f defined on   E  ∖  A is measurable.Now I have a theorem that statesFor a measurable subset   D of   E, f is measurable on   E if and only if   f      ∣    D   and   f      ∣          E      ∖      D       are both measurable.  f      ∣          E      ∖      A       being measurable follows from the fact that   f is continuous on   E∖  A, but how can I show that   f is measurable on   A? I only know that the measure of   A is zero, but nothing about the behavior of   f there?

Sydnee Villegas · Accepted Answer

Actually f is Borel measurable.Hints: We may as well take the domain to be the whole real line (by looking at   f      χ    D  ). Let       f    n    (  x  )  =  f  (            [      n      x      ]        n    ). Check that each       f    n   is Borel measurable. Let   g  (  x  )  =  lim  sup      f    n    (  x  ). Then g is Borel measurable and f=g except at finite number of points. Can you finish? [Inverse images of a Borel set under f and g differ only by a finite