Analyticity on translates of a Jordan curve

Let $\Omega$ be a domain in $\mathbb{C}$ which is symmetric with respect to the real axis and whose boundary is a real analytic simple closed curve. Translate $\overline\Omega$ vertically to get $K = \bigcup\{ \overline\Omega +it, -r\leq t\leq r\}$ where $r>0$ is such that $(\overline\Omega -ir)\cap(\overline\Omega +ir)= \emptyset$. We prove that if $f$ is a continuous function on $K$ such that for each $t, -r\leq t\leq r$, the function $f\vert(b\Omega+it)$ has a continuous extension to $\overline\Omega +it$ which is holomorphic on $\Omega +it$, then $f$ is holomorphic on ${\rm Int}K$.