Are every Riemann integrable functions can be expressed as a linear combination of a finite number of monotone functions?

Question

If $ f:[a,b]\to\mathbb{R} $ is any Riemann integrable function on $ [a,b] $. Can we express $ f $ as a linear combination of a finite number of monotone functions $\phi_j$? $$ f=\sum\limits_{j=1}^{N} \phi_j $$ where $\phi_j$ is monotone function on $(\xi_{j-1},\xi_{j})$ and $a=\xi_0<\xi_1<\cdots<\xi_N=b$.

Answer 1

No. Montone functions are differentiable almost everywhere. If $f$ is a continuous nowhere differentiable function then $f$ is Riemann integrable but it cannot be expressed as a finite sum of monotone functions.

Answer 2

Another argument goes like this. Monotone functions have countable number of discontinuities and hence their linear combination also has a countable number of discontinuities.

However there are Riemann integrable functions which have uncountable number of discontinuities eg the indicator function of Cantor set (of measure zero).