Changing the values of a function $f:[a,b] \to \mathbb R$ of bounded variation for countably infinitely many points not a dense subset of $[a,b]$

Question

Let $f:[a,b] \to \mathbb R$ be a function of bounded variation. It is known that if we change its values at finitely many points of $[a,b]$, then the changed function still remains of bounded variation on $[a,b]$. My question is, suppose we change its values at countably infinitely many points of $[a,b]$ such that the set of this points is not dense in $[a,b]$; then is it true that this changed function is of bounded variation?

Answer 1

The answer is no. For example, if you change values at $\{x_n:n\in\Bbb{N}\}$ to $f(x_n)=n$, and let $\mathcal{P} = \{a<x_{0}, x_{1}, x_{2}\cdots x_{n}\cdots<b\}$ be a partition of $[a, b]$. Then $$ \sum_{i=1}^n|f(x_{i+1})-f(x_{i})|\geqslant\sum_{i=1}^n(|f(x_{i+1})|-|f(x_{i})|)=n $$ So new variation will become unbounded.