Given is a function $f:I \rightarrow \mathbb{R}$ on a closed and bounded interval $I \subset \mathbb{R}$. If $f$ is of bounded variation on $I$, that is $\text{Var}_I(f) < \infty$, then we know that the number of discontinuities of $f$ is at most countable.
My questions:
- Does that also imply that the function is monotone on at most a countable number of subintervals in $I$?
- Does there exist an additional condition that guarantees a finite number of subintervals on which the function is monotone?
Restriction: One important restriction is that I cannot use the number of sign changes of the derivative.
Extra information: In case the above question is too general, then I wonder if such statements can be made about a function that is made up by adding up, composing, or multiplying integer root functions (such as square root and cube root), trigonometric functions and inverse trigonometric functions and where $I \subset \left[0,\pi\right]$.