Can an integrable discontinuous function have vertical asymtpotes?

Question

This is a general question .

I was thinking, as integrable discointinuous functions are indeed improper integrals, which follow to be limits, that vertical asymptotes may exist.

However, i've seen people say it is is impossible whilst others say its possible. I'm pretty confused, any help would be greaT :)

Answer 1

If you are talking about improper integrals, there is no problem. Consider $f\colon(0,1]\longrightarrow\mathbb R$ defined by $f(x)=\frac1{\sqrt x}$. Then\begin{align}\int_0^1f(x)\,\mathrm dx&=\lim_{\varepsilon\to0}\int_\varepsilon^1\frac1{\sqrt x}\,\mathrm dx\\&=\lim_{\varepsilon\to0}\left(2\sqrt1-2\sqrt\varepsilon\right)\\&=2.\end{align}