  We study the continuity of half-plane capacity as a function of boundary
hulls with respect to the Carath\'eodory convergence. In particular, our
interest lies in the case that hulls are unbounded. Under the assumption that
every hull is contained in a fixed hull with finite imaginary part and finite
half-plane capacity, we show that the half-plane capacity is indeed continuous.
We also discuss the extension of this result to the case that the underlying
domain is finitely connected.
