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Let $H^{\infty}$ be the Banach algebra of bounded analytic functions on theunit open disc $\mathbb{D}$ equipped with the supremum norm. As well known,inner functions play an important role of in the study of bounded analyticfunctions. In this paper, we are interested in the study of inner functions.Following by the canonical inner-outer factorization decomposition, define$Q_{inn}$ and $Q_{out}$ the maps from $H^{\infty}$ to $\mathfrak{I}$ the set ofinner functions and $\mathfrak{F}$ the set of outer functions, respectively. Inthis paper, we study the $H^{2}$-norm continuity and $H^{\infty}$-normdiscontinuity of $Q_{inn}$ and $Q_{out}$ on some subsets of $H^{\infty}$. Onthe other hand, the Beurling theorem connects invariant subspaces of themultiplication operator $M_z$ and inner functions. We show the nonexistence ofcontinuous cross section from some certain invariant subspaces to innerfunctions in the supremum norm. The continuity problem of $Q_{inn}$ and$Q_{out}$ on $\textrm{Hol}(\overline{\mathbb{D}})$, the set of all analyticfunctions in the closed unit disk, are also considered.

Continuity of inner-outer factorization and cross sections from invariant subspaces to inner functions