Question: Suppose $(a_i)_{i=1}^\infty$ is a sequence of numbers in [0,1]. Find an absolutely integrable function $f$ on $[0,1]$ with singularities $(a_i)$. Here the integral is in the sense of Riemann integral.
I found a possible answer but I do not know how to examine it when the sequence $(a_i)$ is given in an arbitrary way, or in the special case where $(a_i)$ exhausts all the rationals in $[0,1]$. Can somebody give me a hint on the examination of this function or give me another example if it doesn't work?
My answer: define f(x) to be $$ f(x) = \frac{1}{\limsup_{n \to \infty}\left(\sqrt[2n]{\prod_{i=1}^n|x-a_i|}\right)}$$ if $ x\neq a_i$ and 0 if $ x=a_i$.