I am trying to come up with a function such that
$\int_{-\infty}^{\infty} f(x) dx$ coverges or $\int_{-a}^{a} g(x) dx$ converges where $g(x)$ is not defined on $x = -a$ or $x = a$ (so both will be improper integrals)
The integrand cannot have complex numbers, be $0$ (or some constant, not that it would work), must be real, must also be continuous on $x \in (-a,a)$ (so piecewise functions don't count), and if possible be symmetric (if the function is odd, only look at the infinity domain case)