While i am reading one example in the book, i came across the book teaching me how to evaluate $\int_{-\infty}^{\infty}\dfrac{\sin x}{x}dx$ by using residue theorem.

However, while they say we still construct a semi circle with radius $R$ and centered at $0$, and by letting $\int_{C_R}\dfrac{e^{iz}-1}{z}dz$, we can slowly solve the question.

However, they claimed that $\int_{C_R}\dfrac{e^{iz}-1}{z}dz = 0$ for the closed semi circle $C_R(0,R)$. I have some confusion here as i thought that $z = 0$ is a point where the function is not analytic? I do understand now that if $z=0$ is a removable singularity of the function, then when you integrate, you will get $0$. However, $0$ seems to lie on the boundary of the closed semi circle. So i am not sure if $z=0$ is a singularity or not

Please enlighten

The example is from the book Bak and Newman on application of residue theorem