Some questions about inner functions To avoid trival cases, we assume that $f$ is neither a constant nor a finite Blaschke product.
Two celebrated theorems of Frostman say that $f_a(z)$ is actually a Blaschke product for every $|a|<1$ with the possible exception of a set of logarithmic capacity zero and if $w=f(z)$ is not reducing to a finite Blaschke product or a constant, it will assume infinitely often every value of $|w|<1$ except for a possible set of logarithmic capacity zero. According to above two theorems, there are two exceptional sets associating with each inner function $f(z)$:
${\mathcal E}_1(f)=\{w:|w|<1, f(z)$ assumes $w$ at most finitely often$\}$
and
${\mathcal E}_2(f)=\{w:|w|<1, \frac{f(z)-w}{1-{\overline w}f(z)}$ is not a Blaschke product $\}.$
It is known that the points in ${\mathcal E}_2(f)$ are radial limits of $f$ and the inclusion ${\mathcal E}_1(f)\subseteq{\mathcal E}_2(f)$ always holds. However there maybe also exits points in ${\mathbb D}\setminus{\mathcal E}_2(f)$ are radial limits of $f$. Frostman has constructed the following Blaschke product:
$$B(z)=\prod_{k=1}^{\infty}\frac{(1-\frac{1}{k^2})-z}{1-(1-\frac{1}{k^2})z}$$
which has the radial limit $0$ at $z=1$. We denote these points by ${\mathcal E}_3(f)$.
questions:
Question 1. ${\mathcal E}_2(f)\setminus{\mathcal E}_1(f)$ is at most a denumerable set?
Question 2. ${\mathcal E}_3(f)$ is at most a finite set?
Question 3. If $a\in{\mathcal E}_2(f)\setminus{\mathcal E}_1(f)$, then the singular measure of the singular part of $f_a(z)$ is discrete(i.e., it consists entirely of point masses)? Where $f_a(z):=\frac{f(z)-a}{1-{\overline a}f(z)} (|a|<1)$.
 A: I think that the answer to all three questions is no. The simplest is 3.
Take any inner function $f\neq 1$ with arbitrary non-zero singular measure, and
infinitely many zeros. 
Then $f$ is an not a Blaschke product,
thus $0\in E_2(f)\backslash E_1(f)$, and the singular measure of $f=f_0$
is arbitrary.  
Negative answer to 2 follows from a result of Otsuka (Proc AMS 1954, 533-535) that
there exists an inner function for which every $a$ in the unit disc is a radial limit.
Thus $E_3$ is the whole unit disc, while $E_2$ is always of zero capacity.
A counterexample to 1 that I see at this moment is more difficult. It is based on a theorem of Lehto, Ann. Acad. Sci. Fenn. Ser. A. I. (1954). no. 177, which says
that if $a$ is a radial limit corresponding to a direct singularity 
of $f^{-1}$ then $f_a$
is not a Blaschke product. One can construct $f$ with uncountably many direct singularities,
but such that $E_1$ will be empty. The construction is a bit too complicated
to fit in MO window. And perhaps a simpler example can be found somewhere in Lehto's papers.
