I am having trouble with the following problem:
Let $f:[a,b]\to \mathbb R$ be differentiable on $[a,b]$ and $f^\prime$ is of bounded variation on $[a,b]$. Prove that $f^\prime$ is continuous on $[a,b]$.
My guess is affirmative. I know that $f^\prime$ can be expressed as difference of two monotonic functions $f^\prime=g-h$ on $[a,b]$. Since $f^\prime$ satisfies intermediate value property (i.e. for any $\lambda$ such that $f^\prime(x)<\lambda<f^\prime(y)$ then there exists a $t$ in $[a,b]$ between $x$ and $y$ such that $f^\prime(t)=\lambda$). From this I am attempting to show that since $f^\prime=g-h$ and both of $g$ and $h$ are monotone, then $f^\prime$ will be continuous.
But I do not make it. I have seen a nearly similar type of questions in the MSE but I find hard to address it. Let me know whether my guess is right and if so how to proceed further. Thanks for your attention.