Meromorphic and Harmonic Functions Inducing Continuous Maps from MH∞ into the Riemann Sphere

We study the class of maps from the open unit disk into the Riemann sphere or into [−∞, +∞] that can be continuously extended to the maximal ideal space of H∞. Several characterizations are given for these classes and the subclasses of meromorphic and harmonic functions in terms of cluster sets, spherical gradients, and Carleson measures.