Let $f:[a,b] \to \mathbb{R}$ be a function.
Conjecture: $f$ is of bounded variation on $[a,b]$ if and only if for every $x_0\in (a,b)$ there exists $[c,d] \subseteq [a,b]$ such that $x_0 \in(c,d)$ and $f$ is of bounded variation on $[c,d]$.
Is the conjecture true? The forward direction is obvious (just take $[c,d]=[a,b]$), but I don't see how to prove the converse. I know Jordan's Theorem says that a function is of bounded variation iff it can be written as a difference of increasing functions, but I don't see a way to use that here.
You can see that this is not true without some a priori assumption on the behaviour of $f$ at $a$ and $b$ by considering some extension of $f(x)=\sin(1/(x-a))\sin(1/(b-x))$.