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

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 $$I \subset \left[0,\pi\right]$$.

• Why can't you do an analysis on $f'$? That seems (to me) like the most direct method. Commented Jun 14 at 15:22
• @WhileIAm thanks a lot for your comment. I have considered this, one could analyse the number of roots of $f'$ to answer this question. If $f$ just consisted of products and sums of trigonometric functions, this would be easy since then $f'$ will be a trigonometric polynomial, for which we can bound the number of zeros depending on the degree. However, $f$ also involved roots, inverse trigonometric functions and compositions of those. Therefore the derivative of $f$ will not just be a trigonometric polynomial but something much more complex, for which I do not see an easy way to count roots. Commented Jun 14 at 15:32
• Certainly BV functions are usually not so simple as this. The derivative need not exist everywhere. A BV function could be monotone on no interval. Commented Jun 14 at 15:42

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.

• If you have a more specific question about a class of functions (e.g., your functions in the extra information), I would recommend posting another question with the relevant details. Commented Jun 14 at 16:35
• Thanks a lot for your answer, this is a useful example. However, I know that just having finite variance is not enough to ensure piecewise monotonicity on a finite number of subintervals. What I am asking is, on top of having finite variance, what is an additional condition to ensure that at function is piecewise monotone on a finite number of subintervals. Commented Jun 17 at 8:30