True of False: If $f$ is analytic at each point of a closed contour $L$, then $\int_L f(z) \mathrm{d}z = 0$ True of False: If $f$ is analytic at each point of a closed contour $L$, then $\int_L f(z) \mathrm{d}z = 0$. ...--prophetes.ai

True of False: If $f$ is analytic at each point of a closed contour $L$, then $\int_L f(z) \mathrm{d}z = 0$ True of False: If $f$ is analytic at each point of a closed contour $L$, then $\int_L f(z) \mathrm{d}z = 0$. Justify your answer, and find a counter example if the statement is false. I have an intuition that this statement is false, but I'm having trouble of finding a counter example and explain it. Can anyone help me?

Imagine you have a annular region $1<\vert z\vert<3$ and take a function $f\left(z\right)=\frac{1}{z}$ and a contour $\vert z\vert=2$. Then the integral of this function over this contour is equal $2\pi i$ times residuum at 0 and this is certainly not zero. We just used Cauchy residuum theorem.