= Solution
{c}
The two <holomorphic function> <functions> agree throughout their <overlap region> strip, so they define a single <holomorphic function> on the union of their domains, which is the whole complex plane. The resulting <entire function> is bounded outside a sufficiently large disk by the assumptions at infinity. It is bounded on the disk by <continuity> and compactness. <Liouville theorem> therefore makes it constant:
$$
\boxed{w_+=w_-=C.}
$$
Agreement on an open <overlap region> is what permits <analytic continuation>; equality of unspecified boundary values alone would need additional hypotheses.
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