Let $f:\Bbb R\to \Bbb C$ be a continuous map, if $\int_{-\infty}^{\infty}|f(x)|\,dx$ exists, then $\int_{-\infty}^{\infty}f(x)\,dx$ exists, doesn't it?
2026-03-30 16:22:16.1774887736
Does $\int_{-\infty}^{\infty}f(x)\,dx$ exist?
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I am assuming that you are dealing with the Riemann integral here. Yes, that is correct, assuming that the restriction of $f$ to any interval $[a,b]$ is Riemann-integrable. That's a consequence of Cauchy's criterion.