# Baire one functions, characteristic functions of intervals

Do you think you could help me prove that characteristic functions of intervals are Baire one functions?

And is it true that linear combinations of Baire one functions are also Baire one?

For $[0,1]$, consider the polygonal interpolation of $(-n^{-1},0),(0,1),(1,1),(1+n^{-1},0)$ and $f_n=0$ if $x<-n^{-1}$, $x>1+n^{-1}$.

Use a similar idea for open intervals, and a scaling argument for general intervals.

For the second question, pointwise convergence is preserved by linear combinations, and so is continuity.

• Could you tell me a bit more about the interpolation. How exactly do I show that the characteristic function is Baire one?
– Don
Apr 7 '13 at 11:52
• The interpolation gives a continuous function $f_n$ which converges pointwise to the characteristic function of $[0,1]$. This proves by definition that this function is Baire one. Apr 7 '13 at 11:54
• I'm sorry for bothering you like this. Could you show me exactly how it works?
– Don
Apr 7 '13 at 12:44

We can simply prove that if a function $f : \mathbb{R} \rightarrow \mathbb{R}$ has finitely many discontinuities, then $f$ is Baire one.

The fact above can be found in the paper page $4$. Hence, characteristic function of interval is Baire one since it has finitely many discontinuities.

• You linked to a page of senior projects from years 2005-present... Nov 22 '19 at 6:21