The structure of the space ℱ of
all inner functions in the unit disc D = {z : |z| < 1} under the metric topology
induced by the H∞-norm is considered. It is proven that if two inner functions p and
q belong to the same component of ℱ, then the variation of p∕q on each open arc of
∂D (the boundary of D in the complex plane C) where they can be continued
analytically is bounded by a constant C = C(p,q), independent of the arc. This
criterion is used to show that a component of ℱ can contain nothing but Blaschke
products with infinitely many zeroes, exactly one (up to a constant factor)
singular inner function or infinitely many pairwise coprime singular inner
functions.
