Functions of bounded variations with additional properties

Question

I am considering the following property of a function $f:[a,b]\rightarrow {\mathbb R}$: There exists a finite partition $\{x_0,x_1,\ldots,x_n\}$ of $[a,b]$ such that $a=x_0<x_1<x_2<\cdots <x_n=b$ and $f$ is alternately increasing and decreasing on each subinterval. So for example, $f$ is increasing on $[x_i,x_{i+1}]$ if $i$ is even and decreasing if $i$ is odd. Such a function will clearly be of bounded variation. But $f(x)=x^2\sin(1/x)$ on $[0,1]$ is of bounded variation without having this property. I am trying to understand (in not very precise language) "how far" a generic function of bounded variation can be from having this property. In particular, is there some additional simple property we should require of a bounded variation function so that it will have that property? For example, if we relax the condition that the partition of the interval be countable instead of finite, then the above example would satisfy it. Is there an example of a bounded variation function that does not satisfy the property even if we allow countable partitions?