$f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ?

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If $f\in BV[a,b]$ has the intermediate value property , then is it true that $f$ is continuous on $[a,b]$ ? Please help . Thanks in advacne

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Hints:

  1. $f\in BV[a,b]$ is a difference of two (non-strictly) increasing functions.
  2. $f(a+)$, $f(b-)$, and $f(x\pm)$ exist in $\mathbb R$ for every $x\in (a,b)$.
  3. Assume $f$ has a jump discontinuity at $x\in(a,b)$. Could $f$ have the intermediate value property?