Is the statement true or false?
Let $S\subseteq \mathbb C$ be a domain, $f:S\to \mathbb C$ be a real valued function on $S$. Then integral of $f$ over any contour in $S$ is a real number.
My attempt
If $f$ is analytic, I could conclude that f is a constant function using Cauchy Riemann equation. Then, integral over a closed contour will be zero. I tried to give counter examples also. None of them was working.