Montel's Theorem

Let $f(z)$ be an analytic function of $z$, regular in the half-strip $S$ defined by $a<x<b$ and $y>0$. If $f(z)$ is bounded in $S$ and tends to a limit $l$ as $y\to\infty$ for a certain fixed value $\xi$ of $x$ between $a$ and $b$, then $f(z)$ tends to this limit $l$ on every line $x=x_0$ in $S$, and $f(z)\to l$ uniformly for $a+\delta\leq x_0\leq b-\delta$.

See also Vitali's Convergence Theorem

References

Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 170, 1960.