Erin Lozano

2022-07-01

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 $m\left(A\right)=0$. Now as continuous maps are measurable if $E$ is the measurable domain, then $f$ defined on $E\setminus A$ is measurable.
Now I have a theorem that states
For a measurable subset $D$ of $E$, f is measurable on $E$ if and only if $f{\mid }_{D}$ and $f{\mid }_{E\setminus D}$ are both measurable.
$f{\mid }_{E\setminus A}$ 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

Expert

Actually f is Borel measurable.
Hints: We may as well take the domain to be the whole real line (by looking at $f{\chi }_{D}$). Let ${f}_{n}\left(x\right)=f\left(\frac{\left[nx\right]}{n}\right)$. Check that each ${f}_{n}$ is Borel measurable. Let $g\left(x\right)=limsup{f}_{n}\left(x\right)$. 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

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