# Question Mark Function and continued fraction representations

One could defined Minkowki's question mark question by : $$?(x) = a_0 + 2 \sum_{n= 1}^\infty \dfrac{(-1)^{n+1}}{2^{a_0 +\dots +a_k}},$$ with $x = [a_0;a_1,a_2,\dots]$.

Is Minkowski's question mark function (as defined above) independent of the continued fraction representation used to represent $x$ ?

The continued fraction representation $x=[a_0;a_1,a_2,\ldots]$ is unique when $x$ is irrational. If $x$ is rational, there are exactly two continued fraction representations: $x=[a_0;a_1,\cdots,a_n]=[a_0,a_1,\cdots,(a_{n}-1),1]$, so it's easy to see the question-mark function is indeed independent of continued fraction representation.