pointwise convergent sequence of bounded variation functions

Question

Let $\{f_k\}$ be a sequence of functions. Given:

1. $f_k$ is a bounded variation function on $[a,b]$ for any $k$
2. sequence $f_k$ converges to $f$ point-wise.

Question: is $f$ necessarily a bounded variation function on $[a,b]$?

Answer 1

Let $a=0,b=1$, $f_n(0)=0$, $f_n(x)=\frac 1 x$ for $x > \frac 1 n$, $n$ for $0<x \leq \frac 1 n$. Let $f(x)=\frac 1 x$ for $x>0$ and $0$ for $x=0$. Then $f$ is not even bounded so it is not of bounded variation. Each $f_n$ is of bounded variation because it is decresing and bounded on $(0,1]$.