I am a statistician by training, but I have lately been reviewing the textbook for the real analysis course I took (A Course in Real Analysis by McDonald and Weiss), and there was something that did not appear to be answered in the textbook that I was hoping to get some clarification on.
The set of functions $F$ is defined as containing all continuous functions and is closed under pointwise limits. Any such function $f \in F$ is called a Borel measurable function. I get that this means that if $f_n \in F$ for all $n$ and $f_n \rightarrow f$ pointwise as $n\rightarrow\infty$, then $f \in F$ as well.
With the definition of a characteristic function $\chi_A$ for $A \in \Re$ being: \begin{equation} \chi_A(x) = \begin{cases} 1 & x \in A \\ 0 & x \notin A \end{cases} \end{equation} the book then goes on to define a Borel set $B \subset \Re$ as any set where $\chi_B \in F$; that is to say that $\chi_B$ is a Borel measurable function.
So here inlies my question which I could not find the answer in the chapter for this concept. If $B=\Re$, then $\chi_\Re(x)=1$ for all $x \in \Re$, which is clearly continuous. Also, if $B = \emptyset$, then $\chi_\emptyset(x)=0$ for all $x \in \Re$, also clearly continuous. Other than these two trivial examples, there is no characteristic function that is continuous for any other $B \subset \Re$.
So does this mean that for any $B \subset \Re$ (that is not itself $\Re$ or $\emptyset$), it is that $B$ being a Borel set imply that $\chi_B$ is automatically the pointwise limit of some sequence of continuous functions?
The book has exercises to show that any countable set is a Borel set or that any open interval (and by extension any open set) is a Borel set. I know how to demonstrate that they are indeed but only using the fact I can define a sequence of functions that converges pointwise to a characteristic function for a countable set or open interval, but this is clearly only using the fact that $F$ is closed under pointwise limits. Is this, more or less, the point?