Geodesic
Alternating sums of elements of continued fractions and the Minkowski question mark function
Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 13-16

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We consider a function $A(t)$ $(0\leq t\leq1)$ related to the Minkowski function $?(t)$. $A(t)$ has properties akin to those of $?(t)$ (in particular it satisfies similar functional equations, is continuous and $A'(t)=0$ almost everywhere with respect to Lebesgue measure). But unlike $?(t)$, the function $A(t)$ is not increasing. In reality it is not monotonic on any subinterval of $[0,1]$. 
@article{ZNSL_2017_458_a1,
     author = {E. P. Golubeva},
     title = {Alternating sums of elements of continued fractions and the {Minkowski} question mark function},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {13--16},
     publisher = {mathdoc},
     volume = {458},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a1/}
}
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E. P. Golubeva. Alternating sums of elements of continued fractions and the Minkowski question mark function. Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 13-16. http://geodesic.mathdoc.fr/item/ZNSL_2017_458_a1/