I am trying to prove a proposition that $BV[a.b]\cap C[a.b]$ equipped with the $||\cdot||_\infty$ is Baire 1 category set, which will tell us that $E=\{f:V(f)=\infty, f\in C[a,b]\}$ is a dense Baire 2 category set in $C[a.b]$.
My attempt: I define $F_n=\{f: V(f)\leq n, f\in C[a,b]\}$, then we know that $\cup_{n=1}^{\infty}F_n=BV[a.b]\cap C[a.b]$. I am trying to show that this is a Baire 1 category set, then we are done. In order to show that, we just need to prove the following:
1.$F_n$ is closed.
2. $F_n$ has no interior point for every n.
I have figured out the second claim by using sawtooth functions, but I have some problems when i try to prove the first claim. We suppose $f_n\rightarrow f$ uniformly, then by the definition and some easy calculation, we know that for every $\epsilon>0$, there exists a $m_0$ such that $V(f)\leq V(f_{m_0})+2n\epsilon$, where $n$ is the number of partition (where $a=x_0\leq x_1\leq \cdots\leq x_{n}=b$). So when n goes larger and larger, we can't give an estimation for $V(f)$, this is why i get confused.
My questions: $F_n$ is closed or not? if so, how to prove that? if not so, how do we prove the proposition at first? Any help will be truly grateful.
Best Answer
The fact that $F_n$ is closed is quite elementary.Take any partition $\{x_0,x_1,...,x_N\}$ of $[0,1]$. If $f_k \in F_n$ for all $k$ then $\sum |f_k(x_i)-f_k(x_{i-1})| \leq n$ for all $k$. If $f_k \to f$ in the sup norm then it converges point-wise so we get $\sum |f(x_i)-f(x_{i-1})| \leq n$. Take sup over all partitions to get $V(f) \leq n$.