# Is every Baire one function define on $\mathbb{R}$ continuous a.e.?

Is every Baire one function continuous a.e.? I guess it has a positive answer. Because

1. $f:\mathbb{R} \to \mathbb{R}$ is a Baire one function iff $f$ continuous everywhere except for meagre set.

2. A subset $A$ of $X$ is meagre that it is negligible. So, we can assume that $A$ has a measure zero (I am not sure in this part).

Therefore, $f:\mathbb{R} \to \mathbb{R}$ is a Baire one function iff the discontinuity set of $f$ has measure zero.

• Not every meagre set has measure zero, and you can't just "assume" it is. – Wojowu Mar 30 '17 at 7:20
• math.stackexchange.com/questions/102482/… – Jonas Meyer Mar 30 '17 at 7:27
• @JonasMeyer. Thanks for the link and the answer – flourence Mar 30 '17 at 7:46
• Note that @Jonas Meyer shows that a Baire one function on $\mathbb R$ can be discontinuous a.e.! – Dave L. Renfro Mar 30 '17 at 14:13
• Some types of meager closed sets of positive measure are called "fat Cantor sets". – DanielWainfleet Mar 30 '17 at 21:48