The only functions that I can think with a integral that is constant when evaluated from $-\infty$ to $\infty$ are functions of the form $ f(x) = \frac{k}{x^2 + a}$, in which case $\dots $
$\int_{-\infty}^{\infty} f(x) = 0.$
Of course we could define a piecewise function to be zero for most of the time, but are there any continuous functions whose integral from negative to positive infinity is a non-zero constant?
$\displaystyle\int_{-\infty}^{\infty}e^{-x^{2}}dx>0$ since $e^{-x^{2}}$ is Schwartz and nowhere nozero.