The function is easily seen to be Borel, since the graph of the function can be defined using only natural number quantifiers. In particular, a number is in the support if and only if there is a last place in its digits where $+$ or $-$ appears, followed eventually by ., but then after this, +, - and . do not appear again. That is,

- $x\in S\iff\exists n_1\exists n_2\gt n_1\forall m\geq n_1$ the $m^{\rm th}$ digit of $x$ in base 13 is neither $+$ nor $-$ nor ., except at $n_1$, where it is either $+$ or $-$ and and $n_2$, where it is .

Any set of reals that is definable using only natural number quantification is Borel, since existential quantification over the naturals corresponds to a countable union and universal quantification corresponds to countable intersection. Such sets lie in the arithmetic hierarchy, which is a very low part of the hyperarithmetic hiearchy, which leads ultimately to the Borel sets.

The same idea shows that the graph of the function, as a set of pairs, is Borel.