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This question is motivated by this another one.

Let be some convergent series of positive terms $\sum_{k=1}^\infty a_k$ and some enumeration $\{x_k\}_{k=1}^\infty$ of $\Bbb Q$. Define for $x\in \Bbb R$, $$f(x)=\sum_{x_k<x}a_k$$ This is an increasing function that it is continuous in $\Bbb R\setminus\Bbb Q$ but in no point of $\Bbb Q$. Then the image of $f$ is a subset of the interval $$I=\left(0,\sum_{k=1}^\infty a_k\right)$$

My question: It is there a way (that is, an algorithm) to find if some $y\in I$ is in the image of $f$, for given sequences $\{a_k\}$ and $\{x_k\}$? To take a (relatively) simple example, take $a_k=2^{1-k}$ and $$\{x_k\}=0,1,-1,\frac12,-\frac12,2,-2,\frac13,-\frac13,3,-3,\frac14,-\frac14,4,-4,\frac23,-\frac23\frac 32,-\frac32,\frac15,...$$

And how is $f(\Bbb R)$ like (in a topological sense)? Is its interior void? Is its measure 0?, etc.

ajotatxe
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    At each rational point $x_k$ there is a jump discontinuity of length $a_k$. The values in the interval $[\sum_{x_n<x_k}a_n, \sum_{x_n \leq x_k}a_n)$ are not attained. The range has measure $|I|-\text{sum of the jumps}=\sum_n a_n - \sum_n a_n= 0$. – conditionalMethod Nov 04 '19 at 20:57
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    In between any two different values $f(x_1),f(x_2)\in f(\mathbb{R})$ there is a jump, since there is a rational number between $x_1$ and $x_2$. Therefore, the range's interior is empty. – conditionalMethod Nov 04 '19 at 21:06
  • By reading the other thread, I assume you mean $$f(x)=\sum_{x_k<x}a_k$$ not $\sum_{a_k<x}a_k$? – Paul Sinclair Nov 05 '19 at 04:32

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