# If $\int\limits_\gamma f(z)dz\in\Bbb R$ does it imply $f(z)\in \Bbb R~\forall z\in\gamma$?

If $$\int\limits_\gamma f(z)dz\in\Bbb R$$ does it imply $$f(z)\in \Bbb R~\forall z\in\gamma$$?

This is from a proof where $$\gamma$$ is a circle and we have $$1={1\over 2\pi}\int\limits_\gamma f(z)dz={1\over 2\pi}\int\limits_\gamma Re~f(z)dz$$ and I’m unsure how the second equality is justified.