Let X and Y be separable Banach spaces.


Let $f:X\rightarrow Y$ be a Baire-1 function, which is the pointwise limit of a sequence of continuous functions $f\_n:X\rightarrow Y$.


Define $E$ as the set of $x$ such that $f\_n(x\_n)\rightarrow f(x)$ fails to hold for some sequence $\{x\_n\}$ approaching x.


Is $E$ empty?


If not, if $\mu$ is a countably additive measure on $X$, defined on the Borel $\sigma$-algebra, is $\mu(E)=0$?



Convergence of sequences for Baire-1 functions
Recall:


**1.** Given $f\_n\in C(X,Y)$ point-wise convergent to $f$, the above set $E$ always contains the discontinuity set of $f$ (starting from any $x\_n\to x$ with $ f(x\_n)\not\to f(x)$ one has for a subsequence $ f\_{k\_n}(x\_n)-f(x\_n) \to0$, and re-naming the sequences one can also assume $ f\_{n}(x\_n)-f(x\_n) \to0$ so $ f\_n(x\_n)\not\to f(x)$).


**2.** The characteristic function of any closed set $C$ of a metric space is always of class Baire-$1$: $f\_n(x):=\big(1-n\,d(x,C)\big)\_+$ is $0$ for all $n$ if $x\in C$, and it is eventually $0$ if $x\notin C$.


**3.** Assuming $X$ *separable and of positive dimension,* for any non-zero finite Borel measure $\mu$ on $X$, there is a closed set with empty interior and $\mu(C)>0$, thus a Baire-$1$ function $\chi\_C$ with discontinuity set $\partial C=C$ of positive measure $\mu(C)$ . (Indeed, since $X$ has at most a countable set of points with positive measure, which has empty interior because $\dim X>0$, and since $X$ is separable, $X$ also has a countable dense set $\{s\_n\}\_{n\in\mathbb N}$ of points of measure $0$; since $\mu$ is finite, for each $n\ge1$ there is a ball $B(s\_n,\epsilon\_n)$ of measure less than $3^{-n}\mu(X)$. So $X\setminus \cup\_{n\in\mathbb N}B(s\_n,\epsilon\_n)$ is a closed set with empty interior and measure $\mu\big(X\setminus \cup\_{n\in\mathbb N}B(s\_n,\epsilon\_n)\big)\ge \mu(X)- 
\sum\_{n\in\mathbb N}\mu(B(s\_n,\epsilon\_n))\ge\mu(X)/2>0\;$ ).


