Defining the class of quasicontinuous functions by \begin{equation} QC=(H^{\infty }+C(\dot{{\mathbb R}}))\cap (\overline{H^{\infty }}+C(\dot{{\mathbb R}})). \end{equation} Where $H^{\infty }$ denotes the closed subalgebra of $L^{\infty }({\mathbb R})$ consisting of non-tangential limits on $\mathbb{R}$ of bounded analytic functions on the upper half plane.
My question is: How does the above definition of quasicontinuos functions, relate to the next definition of the same class of functions?
Let $X$ be a topological space. A real-valued function $f:X \rightarrow \mathbb{R}$ is quasi-continuous at a point $x \in X $ if for any every $\epsilon > 0$ and any open neighborhood $U$ of $x$ there is a non-empty open set $G \subset U$ such that
$$|f(x) - f(y)| < \epsilon \;\;\;\; \forall y \in G $$
Or how does the second definition follow from the first?