Alternating sums of elements of continued fractions and the Minkowski question mark function

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 $[0,1]$.