$\displaystyle f$ is continuous on $\displaystyle \mathbb{C}$ and analytic on $\displaystyle \mathbb{C} \setminus \mathbb{R}$.

Prove that $\displaystyle f$ is an entire function.

I know that I need to apply Morera's Theorem which states that if $\displaystyle f$ is a continuous, complex-valued function defined on an open set $\displaystyle D \in \mathbb{C}$, satisfying

$\displaystyle \oint f(z)dz = 0$

for every closed curve $\displaystyle C$ in $\displaystyle D$, then $\displaystyle f$ must be holomorphic on $\displaystyle D$.

I guess I'm not sure how to show that $\displaystyle f$ satisfies $\displaystyle \oint f(z)dz = 0$. I thought I could use Cauchy's integral theorem to show that the function does satisfy this condition but I only know that $\displaystyle f$ is holomorphic on $\displaystyle \mathbb{C} \setminus \mathbb{R}$. Any help would be much appreciated.