Let $X$ be a Polish space (separable completely metrizable). A function $f:X\rightarrow\mathbb{R}$ is said to be of Baire Class $1$ if one of the followings is true:
$(1)$ For any closed subset $P\subseteq X,$ the restriction of $f$ to $P$ has a point of continuity relative to the topology of $P,$
$(2)$ $f$ is the pointwise limit of some sequence of continuous functions $(f_n),$
$(3):$ the pre-image of any open set is $G_{\delta}$ (countable intersection of open sets).
Question: What are the applications of Baire Class $1$ in other fields (Applied Maths, Statistics, Economics,etc)?
I am trying to collect as many applications of Baire Class $1$ functions as possible to motivate students on why we should study them.