Improper integral of an odd function

Question

would $\int\limits_{-\infty}^\infty f(x)\ dx=0$ always be true if $f$ is an odd function?

Answer 1

This is true provided the integral is interpreted as Cauchy principal value. Since $f$ is odd, $$\int_{-A}^Af(x)\,dx=0 $$ Hence, $$\mathrm{p.v. }\int_{-\infty}^{\infty}f(x)\,dx=\lim_{A\to\infty}\int_{-A}^Af(x)\,dx=0 $$ There's also the implicit assumption that $f$ is integrable in each interval $[-a,a]$. If we are not using principal values, then the improper integral $\int_{-\infty}^{\infty}f(x)\,dx$ might not exist. For example, when $f(x)=x$, as @Bungo commented.