Boundary behavior of analytic functions and Approximation Theory

In this Thesis we deal with problems regarding boundary behavior of analytic functions and approximation theory. We will begin by characterizing the set in which Blaschke products fail to have radial limits but have unrestricted limits on its complement. We will then proceed and solve several cases of an open problem posed in \cite{Da}. The goal of the problem is to unify two known theorems to create a stronger theorem; in particular we want to find necessary and sufficient conditions on sets $E_1\subset E_2$ of the unit circle such that there exists a bounded analytic function that fails to have radial limits exactly on $E_1$, but has unrestricted limits exactly on the complement of $E_2$. One of the several cases extends the main theorem proven by Peter Colwell found in [10] regarding boundary behavior of Blaschke products. Additionally, we will provide a shorter proof for the necessity part of the main result in [10] which relies on a classical result proven by R. Baire. The sufficiency part of that result will then be used to shorten another proof by A.J Lohwater and G. Piranian found in [23]. Lastly, we will provide an extension of a well known theorem of Arakeljan about approximating continuous functions which are analytic in the interior of a closed set, by functions analytic in a larger domain.