application of the Cauchy-Goursat Theorem

Question

I'm trying to prove, using the Cauchy-Goursat Theorem, that if a function $f$ is continuous on $\mathbb{C}$ and analytic on every point not on the real axis, then $f$ is analytic everywhere.

I'm not sure where to start. Can anyone help?

Thanks

Answer 1

Hint:

Consider a closed path $\gamma$ that is crossed by the real axis. Now, for each natural $n$, consider two closed paths $\alpha_n$ and $\beta_n$. Each $\alpha_n$ has the same image as $\gamma$ in the upper half of the plane, above $y=1/n$. Similarly $\beta_n$ is under $y=-1/n$. Complete with horizontal segments to close the paths. Prove that $$\int_\gamma f(t)dt=\lim_{n\to \infty}\left(\int_{\alpha_n}f(t)dt+\int_{\beta_n}f(t)dt\right)=0$$