If $S$ is any linear space which contains all real valued monotone functions on $[a,b]$ , then $S$ contains all functions of bounded variation

Question

Let $[a,b]$ be a closed bounded interval in real line and $S$ be any set containing all real valued monotone functions on $[a,b]$ such that $f,g \in S \implies f+g \in S$ and $f \in S , c \in \mathbb R \implies cf \in S$ . Then is it true that $S$ contains all functions of bounded variation o n $[a,b]$ ?

Answer 1

If $f$ has bounded variation and $V$ is the variation of $f$, then $V$ is nondecreasing, $f - V$ is nonincreasing, and $f = V + (f-V)$.