When is function of bounded variation piecewise monotone on finite number of subintervals?

Question

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:

1. Does that also imply that the function is monotone on at most a countable number of subintervals in $I$?
2. 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 $I \subset \left[0,\pi\right]$.

Answer 1

This is an answer to your question (2.). In general, a BV function is not guaranteed to be monotone on a finite number of subintervals. If $f(x) = x^2\sin(1/x)$, then $f' \in L^1[0,1]$, so $f$ is BV, but in every neighborhood of $0$ we have that $f'$ changes sign infinitely many times.

For your question (1.), if I am interpreting your question correctly I believe the answer is yes, as any infinite collection of subintervals contained inside a closed and bounded interval must be at most countable.