Almost Cauchy-Goursat theorem

Let $\displaystyle C$ be the positive oriented frontier of the semi disk $\displaystyle 0\leq r \leq 1, 0\leq \theta \leq \pi$ and let $\displaystyle f(z)$ be a continuous function defined in the semi disk, given by $\displaystyle f(0)=0$ and $\displaystyle f(z)=z^{\frac{1}{2}}= \sqrt r e^{\frac{i\theta}{2}}$, $\displaystyle (r>0, -\frac{\pi}{2}< \theta < \frac{3\pi}{2})$.

Prove that $\displaystyle \int _C f(z)dz=0$.

Is the C-G theorem applicable?

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My attempt : I notice I can't apply C-G theorem because $\displaystyle f(z)$ is not analytic in $\displaystyle z=0$ which is a point in $\displaystyle C$.

I've absolutely no idea how to prove what they ask me.

I'd like an idea...